🧠:[<think>Okay, so I need to find the value of λ that makes the three vectors a, b, and c coplanar. Hmm, coplanar vectors... what does that mean again? If vectors are coplanar, they all lie in the same plane, right? So, they don't span a three-dimensional space. I remember that a way to check if three vectors are coplanar is to use the scalar triple product. If the scalar triple product of a, b, and c is zero, then they are coplanar. Yeah, that's the formula: a · (b × c) = 0. So, I need to compute the cross product of vectors b and c first, then take the dot product with vector a, and set that equal to zero. Then solve for λ. Let me write that down step by step.First, vectors are:a = (2, -1, 3)b = (-1, 4, -2)c = (7, 5, λ)Compute the cross product of b and c. The cross product of two vectors (b₁, b₂, b₃) and (c₁, c₂, c₃) is given by the determinant:i j kb₁ b₂ b₃c₁ c₂ c₃So, expanding that determinant:i*(b₂c₃ - b₃c₂) - j*(b₁c₃ - b₃c₁) + k*(b₁c₂ - b₂c₁)Plugging in the components of b and c:b = (-1, 4, -2)c = (7, 5, λ)So, cross product b × c:i*(4*λ - (-2)*5) - j*(-1*λ - (-2)*7) + k*(-1*5 - 4*7)Let me compute each component step by step.First component (i):4*λ - (-2)*5 = 4λ + 10Second component (j):- [ (-1*λ - (-2)*7 ) ] = - [ (-λ + 14) ] = λ - 14Wait, let's double-check that. The formula is -j*(b₁c₃ - b₃c₁). So:b₁c₃ is (-1)*λ = -λb₃c₁ is (-2)*7 = -14So, b₁c₃ - b₃c₁ = -λ - (-14) = -λ +14Then, multiply by -j: -j*(-λ +14) = j*(λ -14)So, the j component is λ -14Third component (k):(-1)*5 - 4*7 = -5 -28 = -33So, the cross product b × c is:(4λ +10, λ -14, -33)Now, compute the dot product of vector a with this cross product. Vector a is (2, -1, 3).So, a · (b × c) = 2*(4λ +10) + (-1)*(λ -14) + 3*(-33)Let's compute each term:First term: 2*(4λ +10) = 8λ +20Second term: -1*(λ -14) = -λ +14Third term: 3*(-33) = -99Adding them all together:8λ +20 - λ +14 -99Combine like terms:(8λ - λ) + (20 +14 -99) = 7λ + (34 -99) = 7λ -65Set this equal to zero for coplanarity:7λ -65 = 0Solving for λ:7λ = 65 => λ = 65/7Wait, 65 divided by 7. Let me check that division. 7*9=63, so 65-63=2. So, 65/7 = 9 and 2/7, which is approximately 9.2857. But since the problem asks for the exact value, it's 65/7.Hmm, let me verify the calculations again to make sure I didn't make a mistake.Starting with cross product:First component (i): 4λ - (-2)*5 = 4λ +10. Correct.Second component (j): - [ (-1)*λ - (-2)*7 ] = - [ -λ +14 ] = λ -14. Yes, right.Third component (k): (-1)*5 -4*7 = -5 -28 = -33. Correct.Then, dot product with a:2*(4λ +10) = 8λ +20-1*(λ -14) = -λ +143*(-33) = -99Total: 8λ +20 -λ +14 -99 = 7λ -65. Yes, that's correct.Therefore, 7λ =65 => λ=65/7. So, the answer should be 65 over 7. Let me check if that reduces, but 65 and 7 have no common factors, so that's the simplified form.Just to confirm, maybe plugging the value back into the scalar triple product to see if it's zero.Compute cross product b × c with λ=65/7:First component: 4*(65/7) +10 = (260/7 +70/7)=330/7Second component: (65/7) -14 = 65/7 -98/7= -33/7Third component: -33Then, dot product with a:2*(330/7) + (-1)*(-33/7) +3*(-33)= 660/7 + 33/7 -99= (660 +33)/7 -99 = 693/7 -99693 divided by 7: 7*99=693, so 693/7=99Thus, 99 -99=0. Perfect, that checks out.Therefore, λ=65/7 is correct.Final Answerboxed{dfrac{65}{7}}</think>Given the vectors (overrightarrow{a} = (2, -1, 3)), (overrightarrow{b} = (-1, 4, -2)), and (overrightarrow{c} = (7, 5, lambda)), we need to find the value of (lambda) such that the vectors are coplanar. Vectors are coplanar if their scalar triple product is zero. The scalar triple product is calculated as (overrightarrow{a} cdot (overrightarrow{b} times overrightarrow{c})).First, compute the cross product (overrightarrow{b} times overrightarrow{c}):[overrightarrow{b} times overrightarrow{c} = begin{vmatrix}mathbf{i} & mathbf{j} & mathbf{k} -1 & 4 & -2 7 & 5 & lambdaend{vmatrix}= mathbf{i} left(4lambda - (-2) cdot 5right) - mathbf{j} left(-1lambda - (-2) cdot 7right) + mathbf{k} left(-1 cdot 5 - 4 cdot 7right)]Calculating each component:- The (mathbf{i}) component: (4lambda + 10)- The (mathbf{j}) component: (-(- lambda + 14) = lambda - 14)- The (mathbf{k}) component: (-5 - 28 = -33)Thus, the cross product is:[overrightarrow{b} times overrightarrow{c} = (4lambda + 10, lambda - 14, -33)]Next, compute the dot product of (overrightarrow{a}) with this result:[overrightarrow{a} cdot (overrightarrow{b} times overrightarrow{c}) = 2(4lambda + 10) + (-1)(lambda - 14) + 3(-33)]Simplifying each term:- (2(4lambda + 10) = 8lambda + 20)- (-1(lambda - 14) = -lambda + 14)- (3(-33) = -99)Combining these terms:[8lambda + 20 - lambda + 14 - 99 = 7lambda - 65]Setting the scalar triple product to zero for coplanarity:[7lambda - 65 = 0 implies 7lambda = 65 implies lambda = frac{65}{7}]Thus, the value of (lambda) is (boxed{dfrac{65}{7}}).,<think>Okay, so I have this problem here where I need to find the value of λ such that the three vectors a, b, and c are coplanar. The vectors given are:a = (2, -1, 3)b = (-1, 4, -2)c = (7, 5, λ)Hmm, coplanar vectors. Let me recall. If three vectors are coplanar, that means they all lie in the same plane. I think the condition for coplanarity is that the scalar triple product of the three vectors should be zero. The scalar triple product is calculated as the dot product of one vector with the cross product of the other two. And if that's zero, the vectors are coplanar. Right?So the scalar triple product [a, b, c] = a · (b × c) = 0. That should be the condition. So I need to compute the cross product of b and c first, then take the dot product with a, and set that equal to zero. Then solve for λ.Let me start by computing the cross product of vectors b and c. The cross product of two vectors (b1, b2, b3) and (c1, c2, c3) is given by the determinant of the matrix with the standard unit vectors i, j, k in the first row, the components of b in the second row, and the components of c in the third row. So:b × c = |i   j   k|         |-1  4  -2|         |7   5   λ|Calculating this determinant:i*(4*λ - (-2)*5) - j*(-1*λ - (-2)*7) + k*(-1*5 - 4*7)Let me compute each component step by step.First component (i):4*λ - (-2)*5 = 4λ + 10Second component (j):- [ (-1)*λ - (-2)*7 ] = - [ -λ + 14 ] = λ - 14Third component (k):(-1)*5 - 4*7 = -5 - 28 = -33So, putting it all together:b × c = (4λ + 10)i + (λ - 14)j - 33kNow, I need to compute the dot product of vector a with this cross product. Vector a is (2, -1, 3). So:a · (b × c) = (2)*(4λ + 10) + (-1)*(λ - 14) + 3*(-33)Let me compute each term:First term: 2*(4λ + 10) = 8λ + 20Second term: -1*(λ - 14) = -λ + 14Third term: 3*(-33) = -99Now, sum all these terms:8λ + 20 - λ + 14 - 99Combine like terms:(8λ - λ) + (20 + 14 - 99) = 7λ + (34 - 99) = 7λ - 65Set this equal to zero because the scalar triple product must be zero for coplanarity:7λ - 65 = 0Solving for λ:7λ = 65λ = 65 / 7Wait, 65 divided by 7. Let me check my calculations again to make sure I didn't make any mistakes.Starting with the cross product:First component (i): 4λ - (-2)*5 = 4λ +10. Correct.Second component (j): - [ (-1)*λ - (-2)*7 ] = - [ -λ +14 ] = λ -14. Correct.Third component (k): (-1)*5 -4*7 = -5 -28 = -33. Correct.Then the dot product with a:2*(4λ +10) = 8λ +20-1*(λ -14) = -λ +143*(-33) = -99Adding together: 8λ +20 -λ +14 -99 = 7λ + (20+14-99)=7λ + (34 -99)=7λ -65. Correct.So equation: 7λ -65 =0 => λ=65/7. 65 divided by 7 is 9 and 2/7, which is approximately 9.2857.Wait, but is 65/7 the correct answer? Let me verify once again.Alternatively, maybe I made a mistake in the cross product.Wait, let's recalculate the cross product step by step.Given b = (-1,4,-2), c=(7,5,λ)Cross product formula:i*(b2*c3 - b3*c2) - j*(b1*c3 - b3*c1) + k*(b1*c2 - b2*c1)So substituting:i*(4*λ - (-2)*5) - j*(-1*λ - (-2)*7) + k*(-1*5 -4*7)Compute each component:i: 4λ +10j component: - [ -λ - (-14) ] Wait, wait. Wait, let's do it again:Original formula for cross product:If vector b is (b1, b2, b3) and vector c is (c1, c2, c3), then:b × c = (b2c3 - b3c2, b3c1 - b1c3, b1c2 - b2c1)Wait, no, that's the formula for cross product components. Wait, actually, the cross product components are:First component (i): b2c3 - b3c2Second component (j): b3c1 - b1c3Third component (k): b1c2 - b2c1Wait, but with a sign change for the j component. Wait, no, let me check:Wait, no, the cross product formula is:Given vectors b = (b1, b2, b3) and c = (c1, c2, c3), the cross product is:(b2c3 - b3c2, b3c1 - b1c3, b1c2 - b2c1)So, in components:i: b2c3 - b3c2j: b3c1 - b1c3k: b1c2 - b2c1So, substituting:i: 4*λ - (-2)*5 = 4λ +10j: (-2)*7 - (-1)*λ = -14 + λk: (-1)*5 -4*7 = -5 -28 = -33Wait a second! Wait, the j component here is different. Wait, the cross product components as per the formula are:i: b2c3 - b3c2 = 4λ - (-2)*5 =4λ +10j: b3c1 - b1c3 = (-2)*7 - (-1)*λ = -14 + λk: b1c2 - b2c1 = (-1)*5 -4*7 = -5 -28 = -33But then when writing the cross product vector, the j component in determinant calculation is with a negative sign. Wait, no, hold on. Wait, the determinant method gives:When expanding the determinant:i*(b2c3 - b3c2) - j*(b1c3 - b3c1) + k*(b1c2 - b2c1)So the j component is -(b1c3 - b3c1) = -b1c3 + b3c1 = b3c1 - b1c3Which matches the formula above. So the cross product is:(4λ +10, (-2)*7 - (-1)*λ, -5 -28) = (4λ +10, -14 + λ, -33)So that's correct.Therefore, the cross product is (4λ +10, λ -14, -33). Wait, no, hold on. The j component is -14 + λ, which is λ -14. So yes, the cross product is (4λ +10, λ -14, -33)Then, the dot product with a = (2, -1, 3):2*(4λ +10) + (-1)*(λ -14) + 3*(-33)Compute:2*(4λ +10) = 8λ +20(-1)*(λ -14) = -λ +143*(-33) = -99Adding together: 8λ +20 -λ +14 -99 = (8λ -λ) + (20 +14 -99) =7λ + (-65)So 7λ -65 =0 => λ=65/7.So 65 divided by 7 is indeed 9.2857..., which is 9 and 2/7. So as a fraction, 65/7 is the simplest form.Therefore, λ=65/7.Wait, but just to make sure, let me check another approach. Maybe using the determinant of the matrix formed by the three vectors.Since three vectors are coplanar if the determinant of the matrix with them as rows (or columns) is zero.So if we form a 3x3 matrix with vectors a, b, c as rows, then the determinant should be zero.So matrix:| 2   -1   3 ||-1   4   -2 ||7   5   λ |Compute determinant:2*(4*λ - (-2)*5) - (-1)*(-1*λ - (-2)*7) + 3*(-1*5 -4*7)Wait, this is the same as the scalar triple product, so the determinant should be equal to a · (b × c), which we already calculated as 7λ -65. So yes, same result.Therefore, determinant is 7λ -65 =0 => λ=65/7. So the answer is 65/7.Therefore, the value of λ is 65/7.Final AnswerThe value of (lambda) is boxed{dfrac{65}{7}}.</think>Given the vectors (overrightarrow{a} = (2, -1, 3)), (overrightarrow{b} = (-1, 4, -2)), and (overrightarrow{c} = (7, 5, lambda)), we need to find the value of (lambda) such that the vectors are coplanar. Three vectors are coplanar if their scalar triple product is zero. The scalar triple product is calculated as (overrightarrow{a} cdot (overrightarrow{b} times overrightarrow{c})).First, we compute the cross product (overrightarrow{b} times overrightarrow{c}):[overrightarrow{b} times overrightarrow{c} = begin{vmatrix}mathbf{i} & mathbf{j} & mathbf{k} -1 & 4 & -2 7 & 5 & lambdaend{vmatrix}= mathbf{i} (4lambda - (-2) cdot 5) - mathbf{j} (-1lambda - (-2) cdot 7) + mathbf{k} (-1 cdot 5 - 4 cdot 7)]Calculating each component:- The (mathbf{i}) component: (4lambda + 10)- The (mathbf{j}) component: (-(- lambda + 14) = lambda - 14)- The (mathbf{k}) component: (-5 - 28 = -33)Thus, the cross product is:[overrightarrow{b} times overrightarrow{c} = (4lambda + 10, lambda - 14, -33)]Next, we compute the dot product of (overrightarrow{a}) with this result:[overrightarrow{a} cdot (overrightarrow{b} times overrightarrow{c}) = (2) cdot (4lambda + 10) + (-1) cdot (lambda - 14) + (3) cdot (-33)]Calculating each term:- (2 cdot (4lambda + 10) = 8lambda + 20)- (-1 cdot (lambda - 14) = -lambda + 14)- (3 cdot (-33) = -99)Summing these terms:[8lambda + 20 - lambda + 14 - 99 = 7lambda - 65]Setting this equal to zero for coplanarity:[7lambda - 65 = 0 implies lambda = frac{65}{7}]Thus, the value of (lambda) is (boxed{dfrac{65}{7}}).]

🔑:Since vectors overrightarrow{a}, overrightarrow{b}, and overrightarrow{c} are coplanar, there exist scalars p and q such that overrightarrow{c} = poverrightarrow{a} + qoverrightarrow{b}. This can be expressed as the following system of equations:begin{align*}2p - q &= 7, -p + 4q &= 5, 3p - 2q &= lambda.end{align*}To find p and q, let's solve the system for the first two equations:From the first equation, we isolate p:begin{align*}2p - q &= 7, 2p &= 7 + q, p &= frac{7+q}{2}. quadquad (1)end{align*}Now, we substitute p from (1) into the second equation:begin{align*}-left(frac{7+q}{2}right) + 4q &= 5, -frac{7}{2} - frac{q}{2} + 4q &= 5, -frac{7}{2} + frac{7q}{2} &= 5, 7q &= 5 + frac{7}{2}, 7q &= frac{10 + 7}{2}, 7q &= frac{17}{2}, q &= frac{17}{14}, q &= frac{17}{7}. quadquad (2)end{align*}Substituting q back into equation (1) to find p:begin{align*}p &= frac{7 + frac{17}{7}}{2}, p &= frac{frac{49 + 17}{7}}{2}, p &= frac{66}{14}, p &= frac{33}{7}. quadquad (3)end{align*}Having both p and q, substitute them into the third equation to find lambda:begin{align*}lambda &= 3p - 2q, lambda &= 3left(frac{33}{7}right) - 2left(frac{17}{7}right), lambda &= frac{99}{7} - frac{34}{7}, lambda &= frac{65}{7}. quadquad (4)end{align*}Therefore, the value of lambda that ensures the vectors are coplanar is boxed{frac{65}{7}}.

🧠:[<think>Okay, let's see. So the problem is about Weiming Real Estate selling a house to Mr. Qian with a 5% discount, and then three years later, Mr. Qian sells it to Mr. Jin for 60% higher than the original list price. Considering a total inflation rate of 40% over those three years, we need to find the actual profit rate Mr. Qian made, rounded to one decimal place. Hmm.First, let me try to break this down step by step. Maybe if I assign a variable to the original list price, that would help. Let's call the original list price "P". So, Mr. Qian bought the house at a 5% discount. That means he paid 95% of the original price. So his purchase price would be 0.95 * P.Then, three years later, he sold it to Mr. Jin for a price that's 60% higher than the original list price. So the selling price would be the original list price plus 60% of it. That's 1.6 * P. Wait, is that right? If the original price is P, then 60% higher would be P + 0.6P = 1.6P. Yeah, that makes sense.But then we need to consider inflation. The total inflation rate over three years is 40%. So the value of money has decreased by 40%. To calculate the real profit, we need to adjust the selling price and the purchase price for inflation, right?Hmm. So maybe we have to convert both the buying and selling prices into real terms, adjusting for inflation, and then calculate the profit rate based on that. Let me recall how inflation affects profit calculations. Inflation erodes the purchasing power of money, so to find the real profit, we need to account for the fact that the money received in the future is worth less in terms of purchasing power.Alternatively, maybe we can calculate the nominal profit first and then adjust for inflation. Let me think.First, nominal profit. The nominal selling price is 1.6P, and the nominal purchase price was 0.95P. So the nominal profit is 1.6P - 0.95P = 0.65P. Then, the nominal profit rate would be (0.65P / 0.95P) * 100% = (0.65 / 0.95) * 100% ≈ 68.42%. But this is the nominal profit rate, not adjusted for inflation.Since there's been 40% inflation over three years, the real profit rate would be lower. How do we adjust for that? I think we need to calculate the real value of the selling price in terms of the purchasing power at the time of purchase. So, the inflation rate is 40%, which means that the general price level has increased by 40% over three years. Therefore, the real value of the selling price is 1.6P divided by 1.4 (since prices are 1.4 times higher). So the real selling price is (1.6P / 1.4) ≈ 1.14286P.Then, the real purchase price is still 0.95P. So the real profit would be 1.14286P - 0.95P ≈ 0.19286P. Therefore, the real profit rate is (0.19286P / 0.95P) * 100% ≈ (0.19286 / 0.95) * 100% ≈ 20.3%.Wait, but let me check this again. Alternatively, maybe we can use the Fisher equation or something like that. The real rate of return is approximately the nominal rate minus the inflation rate, but that's an approximation. The exact formula is (1 + nominal rate) = (1 + real rate) * (1 + inflation rate). So maybe we can use that.In this case, the nominal rate is 68.42% over three years. Wait, but the inflation rate is 40% over the same period. So, using the exact formula: 1 + real rate = (1 + nominal rate) / (1 + inflation rate). So, that would be (1 + 0.6842) / (1 + 0.4) ≈ 1.6842 / 1.4 ≈ 1.2029. Therefore, the real rate is approximately 20.29%, which rounds to 20.3%. So that matches the previous calculation. So that's good.But let me confirm with another approach. Let's think in terms of purchasing power. When Mr. Qian bought the house, he paid 0.95P. After three years, he sells it for 1.6P. But due to inflation, each unit of currency is worth 1/1.4 times what it was three years ago. So the real value of 1.6P in terms of the original currency's purchasing power is 1.6P / 1.4. Then, subtract the original cost of 0.95P. The real profit is (1.6/1.4)P - 0.95P. Then, divide by the original investment (0.95P) to get the real profit rate.So:Real selling price = 1.6 / 1.4 ≈ 1.142857PReal profit = 1.142857P - 0.95P ≈ 0.192857PProfit rate = 0.192857 / 0.95 ≈ 0.2030, so 20.3%. So yes, this seems consistent.Alternatively, perhaps I can use a numerical example to make this clearer. Let's assume the original list price P is 100. Then, Mr. Qian buys it at 5% discount, so he pays 95.Three years later, he sells it for 60% higher than the original list price, so 100 + 60 = 160.Inflation over three years is 40%, so the purchasing power of money has decreased. To compare the 160 he received in three years to the original 95, we need to adjust the 160 for inflation. Since inflation is 40%, the general price level is 1.4 times higher. Therefore, the real value of 160 in terms of the original dollars is 160 / 1.4 ≈ 114.2857.So, in real terms, he bought the house for 95 and sold it for approximately 114.2857. The real profit is 114.2857 - 95 = 19.2857. The profit rate is (19.2857 / 95) * 100 ≈ 20.3%. That seems to confirm the earlier calculations.Therefore, the answer should be 20.3%, which rounds to 20.3% (already one decimal place). So, 20.3%.Wait, but let me check once again. Let me make sure that the way we account for inflation is correct. So, if inflation is 40% over three years, that means that what cost 1 three years ago now costs 1.40. Therefore, the value of money has decreased. So, 1.40 today is equivalent to 1 three years ago. Therefore, the real value of the selling price is 160 / 1.4 ≈ 114.2857 in terms of three years ago dollars.So, comparing that to the original purchase price of 95, the real profit is 114.2857 - 95 = 19.2857, and 19.2857 / 95 ≈ 0.203, so 20.3%. That seems right.Alternatively, maybe someone might calculate the profit in nominal terms and then adjust for inflation. So, nominal profit is 160 - 95 = 65. Then, adjusted for inflation, that 65 is worth 65 / 1.4 ≈ 46.4286 in real terms. Then, profit rate would be 46.4286 / 95 ≈ 0.4885 or 48.85%? Wait, that contradicts the previous result. Hmm. Wait, that can't be.Wait, maybe this approach is wrong. Because the initial investment was 95, which was already in original dollars. The 160 received is in nominal terms, three years later. So to compare apples to apples, you need to either adjust both to real terms or both to nominal terms.But in the first approach, converting the selling price to real terms (original dollars) and comparing to original cost (already in original dollars). That seems correct.Alternatively, if we convert the original cost to future dollars. So, 95 in original dollars, adjusted for inflation over three years, would be 95 * 1.4 = 133. So the cost in three years' dollars is 133. The selling price is 160. So the nominal profit is 160 - 133 = 27. Then, profit rate is 27 / 133 ≈ 0.203, so 20.3%. So same answer.So this method also gives 20.3%. So that seems to confirm it.Therefore, regardless of whether we adjust the selling price to past dollars or adjust the purchase price to future dollars, the real profit rate is 20.3%. Therefore, the answer is 20.3%, which rounds to one decimal place as 20.3%.Therefore, the answer is boxed{20.3}Final AnswerThe actual profit rate Mr. Qian made is boxed{20.3}%.</think>Let's denote the original list price of the house as ( P ).1. Purchase Price by Mr. Qian: - Mr. Qian bought the house at a 5% discount. - Purchase price: ( 0.95P ).2. Selling Price by Mr. Qian: - Three years later, Mr. Qian sold the house for 60% higher than the original list price. - Selling price: ( 1.6P ).3. Adjusting for Inflation: - The total inflation rate over three years is 40%. - The real value of the selling price in terms of the original purchasing power: [ frac{1.6P}{1.4} approx 1.14286P ]4. Calculating Real Profit: - Real purchase price: ( 0.95P ) - Real selling price: ( 1.14286P ) - Real profit: ( 1.14286P - 0.95P approx 0.19286P )5. Profit Rate Calculation: - Profit rate in terms of the original purchase price: [ left( frac{0.19286P}{0.95P} right) times 100% approx left( frac{0.19286}{0.95} right) times 100% approx 20.3% ]Thus, considering the total inflation rate of 40% over three years, Mr. Qian actually made a profit at a rate of (boxed{20.3})%.,<think>Okay, let's try to figure out this problem step by step. So, Weiming Real Estate Company sold a house to Mr. Qian with a 5% discount off the list price. Then, three years later, Mr. Qian sold it to Mr. Jin for 60% higher than the original list price. But we also have to consider a total inflation rate of 40% over those three years. The question is asking for the actual profit rate Mr. Qian made, adjusted for inflation, right?First, let me break down the information. Let's assume the original list price of the house is some amount. Since the problem doesn't specify, it's probably easiest to just take 100 as the original price. That way, percentages will be easier to work with. So, original list price = 100.Mr. Qian bought it at a 5% discount. So, the discount is 5% of 100, which is 5. Therefore, Mr. Qian paid 100 - 5 = 95 for the house. Got that.Three years later, he sold it to Mr. Jin at a price 60% higher than the original list price. The original list price was 100, so 60% higher than that would be 100 + 60% of 100 = 160. So, he sold it for 160.Now, without considering inflation, his profit would be the selling price minus the buying price. So, 160 - 95 = 65. To find the profit rate, we take the profit divided by the cost. So, 65 / 95 ≈ 0.6842, which is about 68.4%. But wait, the problem mentions considering the total inflation rate of 40% over three years. So, we need to adjust for inflation to find the real profit, not the nominal profit.Inflation reduces the purchasing power of money. So, the 160 he received three years later is in nominal terms, but we need to adjust it to real terms to see the actual increase in wealth. Similarly, maybe the original cost should also be adjusted for inflation? Hmm, let's think.Inflation is 40% over three years. That means the general price level has increased by 40%. So, to compare the purchasing power of the selling price to the purchasing power when he bought it, we need to adjust the selling price for inflation.Alternatively, another way to calculate real profit is to adjust both the selling price and the cost price into the same purchasing power terms, then compute the profit.Let me recall that real rate of return is approximately the nominal rate minus inflation. But in reality, it's (1 + nominal)/(1 + inflation) - 1. So, maybe we can compute the nominal profit rate, then adjust it by inflation.Wait, but here, maybe it's better to adjust the selling price for inflation to see its real value at the time of purchase, then subtract the original cost to find real profit.Alternatively, adjust the original cost for inflation to the time of selling, then compare to the selling price. Both methods should give the same result.Let me try both ways.First method:Selling price is 160 three years later. Inflation is 40% over three years. To find the real value of 160 in terms of the purchasing power at the time of purchase, we divide by 1.40. So, real selling price = 160 / 1.40 ≈ 114.2857.Original cost was 95. So, real profit is 114.2857 - 95 ≈ 19.2857. Then, real profit rate is 19.2857 / 95 ≈ 0.203, or 20.3%.Second method:Original cost was 95. If we adjust this for inflation over three years (40%), the cost in terms of the purchasing power at the time of sale would be 95 * 1.40 = 133. Then, selling price is 160. So, real profit is 160 - 133 = 27. Then, real profit rate is 27 / 133 ≈ 0.203, or 20.3%. Same result.So, either way, the real profit rate is approximately 20.3%. Rounded to one decimal place, that's 20.3%.Wait, but the problem says "Considering the total inflation rate of 40% over three years, Mr. Qian actually made a profit at a rate of _%". So, the answer is 20.3%, so 20.3 rounded to one decimal is 20.3, which would be written as 20.3%. Wait, but let me check if my calculation is correct.Let me do the first method again:Selling price: 160. Inflation is 40%, so real value in purchase year terms: 160 / 1.4 ≈ 114.2857.Cost was 95. So, real profit: 114.2857 - 95 = 19.2857.Profit rate: 19.2857 / 95 = 0.2030, so 20.3%.Alternatively, using the second method:Original cost: 95. Inflated cost after three years: 95 * 1.4 = 133. Selling price is 160. Profit: 160 - 133 = 27. Profit rate: 27 / 95 ≈ 0.2842, which is 28.42%. Wait, no, wait. Wait, if we adjust the cost for inflation to the time of sale, the profit is 27, but the profit rate is calculated relative to the original cost adjusted for inflation? Or relative to the original cost?Wait, this is the confusion. Let me think carefully.When calculating real profit rate, we need to adjust for inflation in such a way that both the cost and the selling price are in the same terms.Either:1. Convert everything to Year 0 (purchase year) currency. Then, selling price is 160 / 1.4 ≈ 114.2857, cost is 95. Profit is 114.2857 - 95 = 19.2857. Profit rate is 19.2857 / 95 ≈ 20.3%.Or,2. Convert everything to Year 3 (sale year) currency. Then, original cost is 95 * 1.4 = 133. Selling price is 160. Profit is 160 - 133 = 27. Profit rate is 27 / 133 ≈ 20.3%.Wait, so in the second method, the profit rate is 27 / 133 ≈ 0.203. So, same as before. So regardless of which way you adjust, the real profit rate is 20.3%.Therefore, the answer should be 20.3%, which rounds to 20.3% with one decimal place. So, 20.3% is already to one decimal place.Wait, but let me check once again to make sure.Original price: 100.Purchase price: 95.Selling price: 160.Inflation: 40% over three years.Compute real selling price: 160 / 1.4 ≈ 114.2857.Original cost in real terms is still 95. So, real profit: 114.2857 - 95 ≈ 19.2857. Profit rate: 19.2857 / 95 ≈ 0.203, so 20.3%.Alternatively, nominal profit: 160 - 95 = 65. Nominal profit rate: 65 / 95 ≈ 68.42%. Then, inflation rate is 40%, so real profit rate is (1 + 0.6842) / (1 + 0.4) - 1 ≈ (1.6842 / 1.4) - 1 ≈ 1.203 - 1 = 0.203, so 20.3%. Same result.So, that's consistent. Therefore, the answer should be 20.3%.But let me check if maybe there's a different interpretation. For example, maybe the question is asking for the real rate of return per year? But the problem states "total inflation rate of 40% over three years", so total, not annual. So, total inflation over three years is 40%, which is the same as 40% total, not compounded annually. Similarly, the selling price is 60% higher than original list price after three years. So, both the profit and inflation are total over the three years, not annual rates.Therefore, the calculation as done before is correct. So, the answer is 20.3%.Wait, but let's verify with exact numbers. Suppose original price is 100.Purchase price: 95.Selling price: 160.Inflation: 40% over three years. So, the inflation factor is 1.4.Real selling price: 160 / 1.4 = 114.2857...Cost was 95. Real profit: 114.2857 - 95 = 19.2857.Profit rate: 19.2857 / 95 = 0.2030, which is 20.3%.Yes, exactly. So, the answer is 20.3%. Rounded to one decimal place, it remains 20.3%.Alternatively, if the problem had specified to use another method, but given the way the question is phrased, adjusting for inflation over the entire period seems correct.Therefore, the answer should be 20.3%.Final AnswerThe actual profit rate Mr. Qian made is boxed{20.3}.</think>Weiming Real Estate Company sold a house to Mr. Qian at a 5% discount off the list price. Three years later, Mr. Qian sold the house to Mr. Jin at a price 60% higher than the original list price. Considering the total inflation rate of 40% over three years, we need to determine Mr. Qian's actual profit rate.1. Original list price: Assume the original list price is 100.2. Purchase price by Mr. Qian: 5% discount on 100, which is (100 - 5 = 95).3. Selling price by Mr. Qian: 60% higher than the original list price, which is (100 + 60 = 160).4. Inflation adjustment: Total inflation rate is 40% over three years. Adjust the selling price for inflation to find its real value in terms of the purchasing power at the time of purchase: [ text{Real selling price} = frac{160}{1.40} approx 114.2857 ]5. Real profit: Subtract the original cost from the real selling price: [ text{Real profit} = 114.2857 - 95 approx 19.2857 ]6. Profit rate: Calculate the profit rate relative to the original cost: [ text{Profit rate} = frac{19.2857}{95} approx 0.203 text{ or } 20.3% ]Thus, considering the total inflation rate of 40% over three years, Mr. Qian actually made a profit rate of (boxed{20.3}).]

🔑:Answer: Mr. Qian's expenditure on purchasing the house was 95% of the list price. Taking into account the inflation factor, the rate of profit Mr. Qian made from selling the house is calculated as follows:frac{1 + 60%}{95% times (1 + 40%)} - 1 = frac{1.6}{0.95 times 1.4} - 1 approx 0.203 = 20.3%.Therefore, the answer is boxed{20.3}.

🧠:[<think>Okay, so I need to find the value of f(5) for this piecewise function. Let me start by understanding the function definition. The function f(x) is defined in two parts: when x is greater than or equal to 10, it's just x minus 2. But if x is less than 10, then f(x) is equal to f(f(x + 6)). Hmm, that seems recursive. Since 5 is less than 10, I have to use the second case. Let me write that down step by step.Starting with f(5). Since 5 < 10, according to the definition, f(5) = f(f(5 + 6)). Let's compute the inner part first. 5 + 6 is 11, so f(5) = f(f(11)). Now, 11 is greater than or equal to 10, so f(11) should be 11 - 2 = 9. Therefore, f(5) = f(9). But wait, 9 is still less than 10, so I need to compute f(9). Using the second case again, f(9) = f(f(9 + 6)) = f(f(15)). Now, 15 is greater than or equal to 10, so f(15) = 15 - 2 = 13. Therefore, f(9) = f(13). Next, compute f(13). Since 13 is >=10, f(13) = 13 - 2 = 11. So going back, f(9) = f(13) = 11. Therefore, f(5) = f(9) = 11. Wait, but let me check again to make sure I didn't make a mistake.Starting over:f(5) = f(f(11))f(11) = 11 - 2 = 9So f(5) = f(9)f(9) = f(f(15))f(15) = 15 - 2 = 13Thus, f(9) = f(13)f(13) = 13 - 2 = 11Therefore, f(9) = 11, so f(5) = 11. Hmm, that's answer B. But let me see if there's another step I might have missed.Wait, maybe I need to check if the recursion continues further. Let me try expanding all steps:First iteration:f(5) = f(f(11)) = f(9) because f(11)=9Then f(9) = f(f(15))= f(13) because f(15)=13Then f(13) = 11So f(9)=11, hence f(5)=11.Alternatively, is there a pattern here? Maybe for x < 10, each time we add 6 until we reach a value >=10, then subtract 2, and maybe subtract 2 again?Wait, let me try to generalize. Let's take x < 10. Then f(x) = f(f(x + 6)). So if x + 6 >=10, then f(x + 6) = (x +6) -2 = x +4. Then f(f(x +6)) = f(x +4). If x +4 is still less than 10, then we have to do this again. So perhaps there's a chain here.Let me try with x=5:x=5: f(5) = f(f(11)) = f(9) [since f(11)=9]x=9: f(9)=f(f(15))=f(13)=11So yes, that gives f(5)=11.Alternatively, maybe if we keep adding 6 until we get to a value >=10, then subtract 2 each time we had to add 6. Let's see:For x=5, how many times do we add 6 before reaching >=10?5 +6 =11 (1 time). Then subtract 2: 11-2=9. Then since 9 is less than 10, add 6 again: 9+6=15 (second time). Subtract 2: 15-2=13. Then since 13 is >=10, subtract 2: 13-2=11. Wait, this seems different. Wait, but this approach might not be the same as the function definition.Wait, perhaps this is a different way to look at it. Let's see.If I try to compute f(5):Since 5 <10, f(5)=f(f(5+6))=f(f(11))Since 11 >=10, f(11)=11-2=9. So f(5)=f(9)Since 9 <10, f(9)=f(f(9+6))=f(f(15))Since 15 >=10, f(15)=15-2=13, so f(9)=f(13)Since 13 >=10, f(13)=13-2=11, so f(9)=11. Therefore, f(5)=11.Yes, this gives the same result. So answer B:11.But wait, let's check another value to see if this pattern holds. Let's take x=8.f(8) = f(f(14)) = f(12) [since f(14)=14-2=12]f(12)=12-2=10, so f(8)=10.But if I follow the alternate approach: 8 +6=14, subtract 2=12. 12 is >=10, so subtract 2=10. So f(8)=10. That works.Alternatively, another example: x=3.f(3)=f(f(9))=f(f(9)).f(9)=f(f(15))=f(13)=11. So f(3)=f(11)=9.Alternatively, according to the "add 6 until >=10, subtract 2 each time":3+6=9 (still <10), add 6 again=15. So two times adding 6. Subtract 2 twice: 15-2-2=11. Wait, but that's not the case. Wait, original approach:f(3)=f(f(9)).f(9)=f(f(15))=f(13)=11.Therefore, f(3)=f(11)=9. Which is different. So maybe the subtract 2 for each addition? Wait, no.Wait, maybe there's another way to look at it. Let me think.Suppose x <10. Then f(x) = f(f(x+6)).If x+6 >=10, then f(x+6)=x+4, so f(x) = f(x+4). If x+4 is still less than 10, then we have to do f(x) = f(f(x+4)) = f(f((x+4)+6)) = f(f(x+10)). But x+10 is definitely >=10 if x is at least 0. Wait, but this might not necessarily be the case. Let me see.Wait, let's take x=5. Then f(5) = f(f(11))=f(9). Then f(9)=f(f(15))=f(13). Then f(13)=11.So, f(5)=11. Similarly, f(9)=11, f(13)=11. Wait, so once you get to 13, you get 11. But 11 is >=10, so f(11)=9. Then f(9)=11 again. Wait, so maybe there's a cycle here between 9 and 11?Wait, f(11)=9, then f(9)=11. So f(f(11))=f(9)=11. Therefore, if you have f(f(x)) where x leads to this cycle, you end up with 11. But in the case of x=5, we have f(5)=f(f(11))=f(9)=11.Similarly, if we take x=7:f(7)=f(f(13))=f(11)=9Then f(7)=9.Wait, but 7+6=13, f(13)=11, f(11)=9. So f(7)=9. Then f(9)=11, so f(f(7))=f(9)=11, and so on.So depending on how many steps you take, you can end up in a cycle between 9 and 11. But in the case of f(5), it goes f(5)=f(f(11))=f(9)=11.So for even number of steps, maybe you end up at 11, and for odd, at 9? Wait, maybe not. Let's see.Wait, in any case, for f(5), following the steps, it's 11. But let me check the answer choices given: A:10, B:11, C:12, D:13. So 11 is option B.But wait, let me see if the problem might have a different approach. Maybe the function is designed such that for x <10, f(x) = x +4, but that might not hold.Wait, if x <10, then f(x) = f(f(x +6)). Let's test x=10, which is the boundary. f(10)=10 -2=8. Wait, but if x=10, it's 8. Then f(8)=f(f(14))=f(12)=10. Then f(10)=8, f(8)=10. Another cycle here. So 8 and 10 cycle. Interesting.Wait, but in any case, the question is about f(5). Following the steps, as above, leads to 11. But let me check once again step by step to make sure.Step 1: Compute f(5). Since 5 <10, use f(x) = f(f(x +6)). So compute x +6 =5+6=11.Step 2: Compute f(11). Since 11 >=10, f(11)=11 -2=9.Step 3: Now, f(5)=f(9). Compute f(9). Since 9 <10, use f(x)=f(f(x +6)). So x +6 =9+6=15.Step 4: Compute f(15). 15 >=10, so f(15)=15 -2=13.Step 5: Now, f(9)=f(13). Compute f(13). 13 >=10, so f(13)=13 -2=11.Step 6: Therefore, f(9)=11. Hence, f(5)=11. So yes, 11 is the answer. Therefore, option B.Alternatively, is there another way to model this function? Maybe working backwards. Let's see.Suppose for x <10, f(x) = f(f(x +6)). Let's denote x +6 as y. If y >=10, then f(y) = y -2. Otherwise, f(y) = f(f(y +6)). So recursively, this might continue until y >=10.So for x <10:If x +6 >=10, then f(x) = f((x +6) -2) = f(x +4).But if x +4 is still less than 10, then f(x +4) = f(f((x +4) +6)) = f(f(x +10)). Since x +10 is definitely >=10 (as x >=0, probably), f(x +10) = (x +10) -2 = x +8. Then f(f(x +10)) = f(x +8). If x +8 >=10, then f(x +8) = (x +8) -2 = x +6. Otherwise, continue.Wait, this seems like a loop. Let's try with x=5:x=5 <10, so f(5)=f(f(11))=f(9). Then x=9 <10, f(9)=f(f(15))=f(13). x=13 >=10, f(13)=11. So f(9)=11, hence f(5)=11. Which matches previous result.Alternatively, if x=4:f(4)=f(f(10))=f(8). Then f(8)=f(f(14))=f(12)=10. So f(4)=10.Wait, so for x=4, f(4)=10. Which is option A. But in our problem, x=5. Hmm. So different x's can lead to different results. So important to follow step by step.Alternatively, is there a formula for f(x) when x <10? Let's try to find a pattern.Take x=9: f(9)=11x=8: f(8)=10x=7: f(7)=f(f(13))=f(11)=9x=6: f(6)=f(f(12))=f(10)=8x=5:11x=4:10x=3: f(3)=f(f(9))=f(11)=9x=2: f(2)=f(f(8))=f(10)=8x=1: f(1)=f(f(7))=f(9)=11x=0: f(0)=f(f(6))=f(8)=10Hmm, so there seems to be a pattern here. For even x <10: x=0:10, x=2:8, x=4:10, x=6:8, x=8:10. Wait, not exactly clear. For odd x <10: x=1:11, x=3:9, x=5:11, x=7:9, x=9:11. So odd numbers less than 10 alternate between 9 and 11. For x=1:11, x=3:9, x=5:11, x=7:9, x=9:11. So starting at x=1 (11), x=3 is 9, x=5 back to 11, and so on. So perhaps depending on how many times you add 6 before reaching a value >=10, and then subtract 2 multiple times.Alternatively, maybe for odd x <10, if (x -1)/2 is even, then f(x)=11, else 9. For example:x=1: (1-1)/2=0 even, so 11x=3: (3-1)/2=1 odd, so 9x=5: (5-1)/2=2 even, so 11x=7:3 odd, 9x=9:4 even, 11That seems to hold. Similarly for even x <10, maybe (x)/2 even or odd determines the result. For example:x=0:0/2=0 even, result 10x=2:2/2=1 odd, result 8x=4:4/2=2 even, result10x=6:6/2=3 odd, result8x=8:8/2=4 even, result10So even x <10: if x/2 is even, result is 10; if x/2 is odd, result is 8.But in any case, for the specific problem of x=5, which is odd, following the pattern, we get 11. Therefore, answer B.But since the problem gives answer choices, and our step-by-step calculation leads to 11, which is option B, I think that's the correct answer. Let me just check once more for x=5.f(5) = f(f(11)) = f(9) because f(11)=9. Then f(9) = f(f(15)) = f(13) =11. So f(5)=11. Yep, that's solid. No mistakes detected in the steps. So the answer should be B:11.Final Answerboxed{B}</think>To find the value of ( f(5) ) for the given piecewise function:[f(x) = begin{cases} x - 2 & text{if } x geq 10 f(f(x + 6)) & text{if } x < 10 end{cases}]we start with ( f(5) ). Since ( 5 < 10 ), we use the recursive case:1. ( f(5) = f(f(5 + 6)) = f(f(11)) )2. Evaluate ( f(11) ). Since ( 11 geq 10 ), ( f(11) = 11 - 2 = 9 ). Thus, ( f(5) = f(9) ).3. Evaluate ( f(9) ). Since ( 9 < 10 ), we use the recursive case again: ( f(9) = f(f(9 + 6)) = f(f(15)) ).4. Evaluate ( f(15) ). Since ( 15 geq 10 ), ( f(15) = 15 - 2 = 13 ). Thus, ( f(9) = f(13) ).5. Evaluate ( f(13) ). Since ( 13 geq 10 ), ( f(13) = 13 - 2 = 11 ). Therefore, ( f(9) = 11 ).Returning to the original call, we have ( f(5) = f(9) = 11 ).Thus, the value of ( f(5) ) is (boxed{B}).,<think>Okay, let me try to figure out how to solve this problem. The function f(x) is defined piecewise: when x is greater than or equal to 10, f(x) is x - 2. But when x is less than 10, f(x) is f(f(x + 6)). The question is asking for f(5), and the options are A through D with numbers 10 to 13. Alright, let's break this down step by step.First, let me understand the function structure. For x >= 10, it's straightforward—subtract 2. But for x < 10, it's recursive. So, if I have to find f(5), since 5 is less than 10, I need to compute f(f(5 + 6)), which simplifies to f(f(11)). Hmm, okay. Let's see. Let me write this down:f(5) = f(f(5 + 6)) = f(f(11))Now, 11 is greater than or equal to 10, so f(11) is 11 - 2 = 9. So now, f(5) becomes f(9). But wait, 9 is still less than 10. So, applying the same rule again:f(9) = f(f(9 + 6)) = f(f(15))15 is greater than or equal to 10, so f(15) = 15 - 2 = 13. Therefore, f(9) = f(13). Now, 13 is also >=10, so f(13) = 13 - 2 = 11. Therefore, f(9) = 11. Going back, f(5) = f(9) = 11. So, the answer would be B: 11. Hmm, but let me check again to make sure I didn't make a mistake.Wait, let me retrace:1. Start with f(5). Since 5 <10, f(5) = f(f(5+6)) = f(f(11))2. Compute f(11). 11 >=10, so f(11)=11-2=93. So now, f(5)=f(9)4. Compute f(9). 9 <10, so f(9)=f(f(9+6))=f(f(15))5. Compute f(15). 15 >=10, so f(15)=15-2=136. Then, f(9)=f(13)7. Compute f(13). 13 >=10, so f(13)=13-2=118. So, f(9)=11, hence f(5)=11That seems consistent. Let me try another angle to verify. Maybe write out each step in more detail.Alternatively, maybe there's a pattern here. Let's try to find a general formula for x <10. Let's suppose x is less than 10. Then, f(x) = f(f(x + 6)). Let's see if we can find a way to express this in terms of x.Suppose we start with some x <10. Then, f(x) = f(f(x + 6)). If x + 6 is still less than 10, then we have to apply the recursive definition again. But in this case, x is 5. x +6 =11 which is >=10, so that's why we could compute f(11) directly. Let me take another example. Let's say x=3. Then f(3)=f(f(9)). Then f(9)=f(f(15))=f(13)=11. So f(3)=f(11)=9. Hmmm. Wait, is this forming some kind of cycle?Alternatively, let's try to compute f(5) again step by step, maybe in a different way. Let's see:f(5) = f(f(11)) = f(9) as before.Then f(9) = f(f(15)) = f(13) = 11.Alternatively, maybe we can write out the recursive calls:f(5) --> f(f(11)) --> f(9) --> f(f(15)) --> f(13) --> 11.So the chain of calls is 5 → f(11) → 9 → f(15) →13 →11. Hence, the result is 11.Alternatively, let's check another possible value. Suppose we need to compute f(4). Then f(4) = f(f(10)). Since 10 is the boundary, f(10) =10 -2=8. So f(4)=f(8). Then f(8)=f(f(14)). f(14)=12. Then f(8)=f(12)=10. So f(4)=10. Wait, is there a pattern here? Let's see:f(5)=11, f(4)=10, maybe f(3)=9? Let's check:f(3)=f(f(9)). f(9)=f(f(15))=f(13)=11. Therefore, f(3)=f(11)=9. So f(3)=9. Then f(2)=f(f(8))=f(10)=8. So f(2)=8. Similarly, f(1)=f(f(7)). f(7)=f(f(13))=f(11)=9. Then f(1)=f(9)=11. Hmm, interesting. So for some values, like x=1, it's 11. For x=3, it's 9. For x=5, it's 11. Maybe the function alternates or has some periodicity?But perhaps I'm overcomplicating. Let's go back to the original problem. The question is f(5). So following through step by step, we get 11. Let's check if the answer is among the options. The options are A:10, B:11, C:12, D:13. So B:11 is an option. Therefore, the answer should be B.Wait a second. Wait, but let me make sure I didn't skip any steps. Let me try another approach. Suppose I unroll the recursion. Let's try to define f(x) for x <10 in terms of f(f(x+6)). Let's see.First, note that if x >=10, then f(x) =x-2. If x <10, then f(x)=f(f(x+6)). So, for x <10, we need to compute f(x+6) first, then compute f of that result. Let's tabulate the values of x from 10 downwards and see if we can find a pattern.Starting with x=10: f(10)=10-2=8x=9: f(9)=f(f(15)). f(15)=13, so f(9)=f(13)=13-2=11x=8: f(8)=f(f(14)). f(14)=12, so f(8)=f(12)=12-2=10x=7: f(7)=f(f(13)). f(13)=11, so f(7)=f(11)=9x=6: f(6)=f(f(12)). f(12)=10, so f(6)=f(10)=8x=5: f(5)=f(f(11)). f(11)=9, so f(5)=f(9)=11x=4: f(4)=f(f(10)). f(10)=8, so f(4)=f(8)=10x=3: f(3)=f(f(9)). f(9)=11, so f(3)=f(11)=9x=2: f(2)=f(f(8)). f(8)=10, so f(2)=f(10)=8x=1: f(1)=f(f(7)). f(7)=9, so f(1)=f(9)=11x=0: f(0)=f(f(6)). f(6)=8, so f(0)=f(8)=10Okay, so creating a table here:x | f(x)---|---10 | 89 | 118 | 107 | 96 | 85 | 114 | 103 | 92 | 81 | 110 | 10Interesting. So for odd numbers less than 10 (except maybe some), f(x) seems to be 11, 9, etc. But looking at the table, for x=5, f(5)=11. So that's consistent with our previous result.Therefore, the answer should be B:11.But let me check once more for x=5:f(5) = f(f(11)) = f(9) [since f(11)=9]Then f(9) = f(f(15)) = f(13) [since f(15)=13]Then f(13)=11 [since 13-2=11]Thus, f(9)=11, so f(5)=11.Yes, that's correct. Therefore, the answer is 11, which is option B.Final Answerboxed{B}</think>To find the value of ( f(5) ) given the piecewise function ( f(x) ):[f(x) = begin{cases} x - 2, & text{if } x geq 10 f(f(x + 6)), & text{if } x < 10 end{cases}]we proceed step by step:1. Calculate ( f(5) ): Since ( 5 < 10 ), we use the recursive definition: [ f(5) = f(f(5 + 6)) = f(f(11)) ]2. Calculate ( f(11) ): Since ( 11 geq 10 ), we use the direct definition: [ f(11) = 11 - 2 = 9 ] Thus, ( f(5) = f(9) ).3. Calculate ( f(9) ): Since ( 9 < 10 ), we use the recursive definition: [ f(9) = f(f(9 + 6)) = f(f(15)) ]4. Calculate ( f(15) ): Since ( 15 geq 10 ), we use the direct definition: [ f(15) = 15 - 2 = 13 ] Thus, ( f(9) = f(13) ).5. Calculate ( f(13) ): Since ( 13 geq 10 ), we use the direct definition: [ f(13) = 13 - 2 = 11 ] Thus, ( f(9) = 11 ).6. Combine the results: Since ( f(9) = 11 ), we have ( f(5) = 11 ).Therefore, the value of ( f(5) ) is (boxed{B}).]

🔑:Analysis: Given f(x)= begin{cases} x-2, & (xgeqslant 10) f[f(x+6)], & (x < 10) end{cases},therefore f(5)=f[f(11)]=f(9)=f[f(15)]=f(13)=11.Thus, the answer is boxed{text{B}}.To find the value of f(5), according to the piecewise function given in the question, we just need to transform the problem into finding the function value when xgeqslant 10.This question primarily tests piecewise functions and finding function values, which is a basic question.

🧠:[<think>Okay, let's see. The problem says that if tanθ + 1/tanθ = 4, then sin2θ is equal to which of the options given. The choices are A through D, with fractions from 1/5 to 1/2. Hmm, I need to find sin2θ given that equation. Alright, let's start by recalling some trigonometric identities.First, I remember that tanθ is sinθ over cosθ. So tanθ + 1/tanθ would be sinθ/cosθ + cosθ/sinθ. Let me write that out:tanθ + 1/tanθ = (sinθ/cosθ) + (cosθ/sinθ).If I add those two fractions together, I need a common denominator, which would be sinθ*cosθ. So:= [sin²θ + cos²θ] / (sinθ*cosθ).Wait, the numerator is sin²θ + cos²θ, which is equal to 1. That's one of the Pythagorean identities. So this simplifies to:1 / (sinθ*cosθ).But the problem states that tanθ + 1/tanθ equals 4. So:1 / (sinθ*cosθ) = 4.Therefore, sinθ*cosθ = 1/4.But wait, I need to find sin2θ. There's an identity for sin2θ which is 2sinθcosθ. Let me write that down:sin2θ = 2sinθcosθ.So if sinθcosθ is 1/4, then multiplying that by 2 gives:sin2θ = 2*(1/4) = 1/2.Wait, but 1/2 is option D. But let me check the options again. The options are A: 1/5, B:1/4, C:1/3, D:1/2. So D is 1/2. But that seems straightforward. Hmm, but let me verify again to make sure I didn't make a mistake.Starting from tanθ + 1/tanθ = 4.Express tanθ as sinθ/cosθ, so we have:sinθ/cosθ + cosθ/sinθ = 4.Combine the fractions:[sin²θ + cos²θ]/(sinθcosθ) = 4.Since sin²θ + cos²θ = 1, so:1/(sinθcosθ) = 4.Therefore, sinθcosθ = 1/4.Then sin2θ = 2sinθcosθ = 2*(1/4) = 1/2.So that's D. But wait, the answer choices don't include 1/2 as a mistake? Let me check the problem again.Wait, the original problem is written as: "If tanθ + 1/tanθ = 4, then sin2θ =". The options are given as A:1/5, B:1/4, C:1/3, D:1/2.So D is indeed 1/2. So according to my calculation, the answer is D. Hmm. But I recall sometimes when dealing with equations involving tangents and sines, there might be cases where the sign matters, depending on the quadrant where θ is located. However, the problem doesn't specify any restrictions on θ, so we can assume that θ is such that tanθ is defined and the expression is valid (i.e., θ ≠ kπ/2 for integer k). Since the given equation is tanθ + 1/tanθ = 4, which is positive, so tanθ must be positive, because 1/tanθ is also positive. So θ is in a quadrant where tangent is positive, which is first or third quadrant. But sine of 2θ: in the first quadrant, 2θ would be in the first or second quadrant, where sine is positive. In the third quadrant, θ is between π and 3π/2, so 2θ is between 2π and 3π, which is equivalent to between 0 and π (since sine has a period of 2π). Wait, 2θ would be between 2π and 3π, which is the same as between 0 and π if we subtract 2π. So sin2θ would still be positive or negative? Wait, sin(2θ) in the third quadrant: if θ is in the third quadrant, then 2θ is in... Let's take θ = π + α, where α is between 0 and π/2. Then 2θ = 2π + 2α, which is coterminal with 2α, which is between 0 and π. So sin2θ would be sin2α, which is positive because 2α is in the first or second quadrant. Similarly, if θ is in the first quadrant, 2θ is in the first or second quadrant, where sine is positive. So regardless of the quadrant, sin2θ is positive. Therefore, the value should be positive 1/2, which matches our previous calculation. So the answer is D:1/2.But let me check if there is another way someone might approach this problem and get a different answer. Maybe squaring both sides or something?Wait, another approach: Let me let x = tanθ. Then the equation becomes x + 1/x = 4. Then, multiply both sides by x to get x² + 1 = 4x. Then, x² -4x +1 =0. Using quadratic formula, x=(4 ±√(16-4))/2 = (4 ±√12)/2 = (4 ±2√3)/2 = 2 ±√3. So tanθ = 2 +√3 or 2 -√3.Hmm, 2 +√3 is approximately 3.732, and 2 -√3 is approximately 0.2679. Both positive, so θ is in first or third quadrant as before. Now, to find sin2θ. Using the identity sin2θ = 2tanθ/(1 + tan²θ). Let me verify that identity. Because sin2θ = 2sinθcosθ, and if we divide numerator and denominator by cos²θ, we get 2tanθ / (1 + tan²θ). Yes, that's correct. So using this identity:If tanθ = x, then sin2θ = 2x/(1 + x²).So let's compute 2x/(1 + x²) for x = 2 +√3.First, compute x = 2 +√3.Then x² = (2 +√3)² = 4 +4√3 +3 = 7 +4√3.So 1 + x² = 8 +4√3.Then 2x = 2*(2 +√3) =4 +2√3.So sin2θ = (4 +2√3)/(8 +4√3). Let's simplify this fraction.Factor numerator and denominator:Numerator: 2*(2 +√3)Denominator: 4*(2 +√3)Therefore, sin2θ = [2*(2 +√3)] / [4*(2 +√3)] = 2/4 = 1/2.Similarly, if x = 2 -√3, then compute sin2θ:x = 2 -√3x² = (2 -√3)² =4 -4√3 +3 =7 -4√31 + x² = 8 -4√32x =4 -2√3So sin2θ = (4 -2√3)/(8 -4√3). Factor numerator and denominator:Numerator: 2*(2 -√3)Denominator: 4*(2 -√3)Thus, sin2θ = [2*(2 -√3)] / [4*(2 -√3)] = 2/4 =1/2.So regardless of which root we take, sin2θ =1/2. Therefore, the answer is D:1/2. So both methods give the same result. Therefore, the correct answer is D.But wait, the options include 1/2, which is D. So why am I doubting? Maybe because the answer seems straightforward? Let me check once more.Alternatively, let's use the first method. Since tanθ + cotθ =4, and we converted that to 1/(sinθcosθ)=4, so sinθcosθ=1/4. Then sin2θ=2*(1/4)=1/2. Yes, that's the same result. So regardless of the method, the answer is 1/2. Therefore, the answer is D. So I think that is correct. Maybe the problem is trying to trick us into thinking it's more complicated, but no, the answer is straightforward.Final Answerboxed{D}</think>Given the equation (tan theta + frac{1}{tan theta} = 4), we need to find (sin 2theta).First, we recognize that (tan theta + frac{1}{tan theta}) can be rewritten using trigonometric identities:[tan theta + frac{1}{tan theta} = frac{sin theta}{cos theta} + frac{cos theta}{sin theta}]Combining these fractions over a common denominator, we get:[frac{sin^2 theta + cos^2 theta}{sin theta cos theta}]Using the Pythagorean identity (sin^2 theta + cos^2 theta = 1), this simplifies to:[frac{1}{sin theta cos theta}]Given that (tan theta + frac{1}{tan theta} = 4), we have:[frac{1}{sin theta cos theta} = 4]Thus, (sin theta cos theta = frac{1}{4}).Next, we use the double-angle identity for sine, (sin 2theta = 2 sin theta cos theta):[sin 2theta = 2 left(frac{1}{4}right) = frac{1}{2}]Therefore, the value of (sin 2theta) is (boxed{D}).,<think>Okay, let's see. The problem says that if tan θ plus 1 over tan θ equals 4, then what is sin 2θ? The options are A through D with fractions. Hmm, I need to remember trigonometric identities here. First, I know that tan θ is the same as sin θ over cos θ. Maybe I can express the given equation in terms of sine and cosine. Let me write that down:tan θ + 1/tan θ = 4So substituting tan θ with sin θ / cos θ:(sin θ / cos θ) + (cos θ / sin θ) = 4Hmm, combining these two terms. To add them together, they need a common denominator. The common denominator would be sin θ cos θ. Let's rewrite each term:[sin^2 θ + cos^2 θ] / (sin θ cos θ) = 4Wait, the numerator is sin squared theta plus cos squared theta, which is equal to 1. So this simplifies to:1 / (sin θ cos θ) = 4So 1 divided by (sin θ cos θ) equals 4. Therefore, sin θ cos θ must equal 1/4. But the question is asking for sin 2θ. I remember that the double angle identity for sine is sin 2θ = 2 sin θ cos θ. So if sin θ cos θ is 1/4, then multiplying by 2 gives 2*(1/4) = 1/2. Wait, so sin 2θ would be 1/2? Looking at the options, D is 1/2. So is the answer D?Wait a second, let me check my steps again to make sure I didn't make a mistake. Starting with tan θ + 1/tan θ = 4. Substitute tan θ as sin/cos:sin/cos + cos/sin = 4. Combine the fractions:(sin^2 + cos^2)/(sin cos) = 4. Since sin^2 + cos^2 = 1, this becomes 1/(sin cos) = 4. So sin cos = 1/4. Then sin 2θ = 2 sin cos = 2*(1/4) = 1/2. That seems right. But let me also think if there's another way to approach this. Maybe using algebraic manipulation with tan θ. Let's let x = tan θ. Then the equation becomes x + 1/x = 4. Multiplying both sides by x gives x^2 + 1 = 4x. Then rearranging: x^2 - 4x + 1 = 0. Solving this quadratic equation for x:x = [4 ± sqrt(16 - 4*1*1)] / 2 = [4 ± sqrt(12)] / 2 = [4 ± 2*sqrt(3)] / 2 = 2 ± sqrt(3)So tan θ is 2 + sqrt(3) or 2 - sqrt(3). Hmm, interesting. But how does this help me find sin 2θ?Alternatively, I recall that tan θ + 1/tan θ can be expressed in terms of sine and cosine as we did before. But maybe using the identity for sin 2θ in terms of tan θ. Let me recall that sin 2θ is 2 tan θ / (1 + tan^2 θ). Let me verify that identity. Yes, since sin 2θ = 2 sin θ cos θ, and if you divide numerator and denominator by cos^2 θ, you get (2 sin θ / cos θ) / (1 + sin^2 θ / cos^2 θ) = 2 tan θ / (1 + tan^2 θ). So that's correct.So if we let t = tan θ, then sin 2θ = 2t / (1 + t^2). But from the given equation, t + 1/t = 4. Let me see if I can relate 2t / (1 + t^2) to this.Alternatively, from t + 1/t = 4, we can square both sides. Let's try that:(t + 1/t)^2 = 16Which gives t^2 + 2*(t)*(1/t) + 1/t^2 = 16Simplify: t^2 + 2 + 1/t^2 = 16So t^2 + 1/t^2 = 14Hmm, not sure if that helps. But maybe if I need to compute sin 2θ, which is 2t/(1 + t^2). Let me denote S = sin 2θ = 2t/(1 + t^2). Let's see if we can express S in terms of t + 1/t.Let me let t + 1/t = 4. Let me write t + 1/t = 4. Let me denote this as equation (1). Also, from equation (1), t^2 + 1/t^2 = 14, as we saw earlier.But how to connect this to S = 2t/(1 + t^2). Let me manipulate S:S = 2t / (1 + t^2). Let's divide numerator and denominator by t:S = 2 / ( (1/t) + t ). But (1/t) + t is 4, from equation (1). So S = 2 / 4 = 1/2. Oh! That's a clever way. So S = 2 / (t + 1/t) = 2/4 = 1/2. So sin 2θ = 1/2. So again, the answer is D: 1/2. Wait, that's the same answer as before. So both methods lead to the same conclusion. Therefore, the answer must be D.But just to make sure, let's try plugging in the value of tan θ we found earlier. If tan θ is 2 + sqrt(3), then what's sin 2θ?Since tan θ = 2 + sqrt(3), then θ is arctan(2 + sqrt(3)). But 2 + sqrt(3) is actually tan(75 degrees), since tan(60 + 15) = tan 75. Using the addition formula, tan(A + B) = (tan A + tan B)/(1 - tan A tan B). Let's check:tan(60° + 15°) = tan 75°, tan 60° = sqrt(3), tan 15° = 2 - sqrt(3). Wait, tan(45° - 30°) = (1 - sqrt(3)/3)/(1 + sqrt(3)/3) = (3 - sqrt(3))/(3 + sqrt(3)) = multiply numerator and denominator by (3 - sqrt(3)): (9 - 6 sqrt(3) + 3)/(9 - 3) = (12 - 6 sqrt(3))/6 = 2 - sqrt(3). So tan 15° is 2 - sqrt(3). Therefore, tan 75° is tan(60° + 15°) which would be (sqrt(3) + (2 - sqrt(3)))/(1 - sqrt(3)*(2 - sqrt(3))). Let's compute numerator: sqrt(3) + 2 - sqrt(3) = 2. Denominator: 1 - sqrt(3)*(2 - sqrt(3)) = 1 - 2 sqrt(3) + 3 = 4 - 2 sqrt(3). So tan 75° = 2 / (4 - 2 sqrt(3)) = multiply numerator and denominator by (4 + 2 sqrt(3)): 2*(4 + 2 sqrt(3)) / (16 - 12) = (8 + 4 sqrt(3))/4 = 2 + sqrt(3). So yes, tan 75° = 2 + sqrt(3). Therefore, θ could be 75°, so 2θ is 150°, and sin 150° is 1/2. Which matches the answer. Similarly, if tan θ = 2 - sqrt(3), then θ is 15°, so 2θ is 30°, and sin 30° is 1/2. So either way, sin 2θ is 1/2. So that confirms it again. Therefore, the correct answer is D: 1/2.Final Answerboxed{D}</think>Given the equation (tan theta + frac{1}{tan theta} = 4), we need to find the value of (sin 2theta).First, we rewrite the given equation in terms of sine and cosine:[tan theta + frac{1}{tan theta} = frac{sin theta}{cos theta} + frac{cos theta}{sin theta} = 4]Combining the fractions, we get:[frac{sin^2 theta + cos^2 theta}{sin theta cos theta} = 4]Using the Pythagorean identity (sin^2 theta + cos^2 theta = 1), this simplifies to:[frac{1}{sin theta cos theta} = 4]Thus, (sin theta cos theta = frac{1}{4}).We use the double angle identity for sine, (sin 2theta = 2 sin theta cos theta):[sin 2theta = 2 left(frac{1}{4}right) = frac{1}{2}]To verify, we can also solve the quadratic equation derived from letting (x = tan theta):[x + frac{1}{x} = 4 implies x^2 - 4x + 1 = 0]Solving this quadratic equation, we find (x = 2 pm sqrt{3}). Using the identity (sin 2theta = frac{2 tan theta}{1 + tan^2 theta}), we substitute (x = 2 pm sqrt{3}) and find that (sin 2theta = frac{1}{2}).Both methods confirm that (sin 2theta = frac{1}{2}).[boxed{D}]]

🔑:Solution: sin 2theta = 2sin theta cos theta = frac {2sin theta cos theta}{sin ^{2}theta +cos ^{2}theta} = frac {2tan theta}{tan ^{2}theta +1} = frac {2}{tan theta + frac {1}{tan theta}}= frac {2}{4}= frac {1}{2} Therefore, the correct option is boxed{D}. First, use the double angle formula for sine to transform, then divide by 1, replace 1 with the trigonometric function relationship of the same angle, and simplify using the method of homogeneous expressions to find the answer. This question mainly tests the double angle formula and the application of homogeneous expressions, as well as computational skills, and is considered a basic question.

🧠:[<think>Okay, let me try to figure out this problem. So there's a magical tree with 63 fruits. On the first day, 1 fruit falls. Then each subsequent day, the number of fruits that fall is one more than the previous day. But if there aren't enough fruits left to drop the required number, it starts over by dropping 1 fruit the next day, and continues increasing again. The question is, on which day will all the fruits have fallen?Hmm, let me break this down. So the falling pattern is supposed to be 1, 2, 3, 4, ... each day, increasing by 1. But if the remaining fruits are less than what's supposed to fall that day, then instead it drops 1 fruit and starts the sequence over. So, for example, if on day 5, it's supposed to drop 5 fruits, but there are only 3 left, then it drops 1 instead, and the next day would be 2, and so on.So we need to simulate this process day by day until all 63 fruits are gone. Let me try to keep track step by step.Start with 63 fruits.Day 1: Drop 1 fruit. Remaining: 63 - 1 = 62.Day 2: Drop 2 fruits. Remaining: 62 - 2 = 60.Day 3: Drop 3 fruits. Remaining: 60 - 3 = 57.Day 4: Drop 4 fruits. Remaining: 57 - 4 = 53.Day 5: Drop 5 fruits. Remaining: 53 - 5 = 48.Day 6: Drop 6 fruits. Remaining: 48 - 6 = 42.Day 7: Drop 7 fruits. Remaining: 42 - 7 = 35.Day 8: Drop 8 fruits. Remaining: 35 - 8 = 27.Day 9: Drop 9 fruits. Remaining: 27 - 9 = 18.Day 10: Drop 10 fruits. Wait, but there are only 18 left. 18 - 10 would be 8, which is okay. But hold on, let me check: 18 remaining, need to drop 10. 18 >= 10, so it's okay. So remaining: 18 - 10 = 8.Day 11: Now, according to the original rule, the next day should be 11 fruits. But we only have 8 left. 8 < 11, so instead, we start over by dropping 1 fruit. So Day 11: Drop 1. Remaining: 8 - 1 = 7.Day 12: Now, since we started over, Day 12 should be 2 fruits. But we have 7 left. 7 >= 2, so drop 2. Remaining: 7 - 2 = 5.Day 13: Next day, 3 fruits. 5 >= 3, so drop 3. Remaining: 5 - 3 = 2.Day 14: Next day, 4 fruits. But only 2 left. So start over. Drop 1. Remaining: 2 - 1 = 1.Day 15: Next day, 2 fruits. But only 1 left. So drop 1. Now, remaining: 1 - 1 = 0.Wait, so on Day 15, the last fruit falls. So the answer is Day 15.Wait, let me verify this step by step again to make sure I didn't make a mistake.Total fruits: 63.Day 1: 1 -> 62 left.Day 2: 2 -> 60 left.Day 3: 3 -> 57 left.Day 4: 4 -> 53 left.Day 5: 5 -> 48 left.Day 6: 6 -> 42 left.Day 7: 7 -> 35 left.Day 8: 8 -> 27 left.Day 9: 9 -> 18 left.Day 10: 10 -> 8 left.Day 11: Can't drop 11, so 1 -> 7 left.Day 12: Next in sequence would be 2 -> 5 left.Day 13: Next is 3 -> 2 left.Day 14: Next is 4, but only 2 left, so drop 1 -> 1 left.Day 15: Next is 2, but only 1 left, so drop 1 -> 0 left.Yes, that seems correct. So total days: 15.But let me check if there's a formula or a smarter way to compute this without enumerating each day. Maybe using the sum of consecutive numbers.The sum of the first n days is n(n+1)/2. If the total sum equals 63, then n(n+1)/2 = 63. Solving this quadratic equation: n² + n - 126 = 0.Using the quadratic formula: n = [-1 ± sqrt(1 + 504)] / 2 = [-1 ± sqrt(505)] / 2. sqrt(505) is approximately 22.47, so positive solution is (21.47)/2 ≈ 10.73. So around 10 or 11 days. But since the sum up to day 10 is 10*11/2 = 55, which is less than 63. Sum up to day 11 is 66, which is more than 63. So if there were no reset, it would take 11 days. But since the tree resets when there's not enough fruits, we have to adjust.But in reality, when we get to day 10, we have 8 fruits left. Then day 11 would need to drop 11, but only 8 left. So reset to 1. Then days 11-15 as above. So the total days are longer.Alternatively, maybe we can think in cycles. The first cycle is days 1-10, dropping 55 fruits (sum 1-10). Then remaining 63 - 55 = 8 fruits. Then starting a new cycle with 1,2,3,... but only 8 left. The next cycle would normally need sum up to 8. Let's see: 1 + 2 + 3 + 4 = 10, which is too much. So 1 + 2 + 3 = 6, which is less than 8. Then remaining 8 - 6 = 2. Then next cycle: 1 + 2 = 3, which is more than 2. So drop 1, then 1. So total days: 10 (first cycle) + 3 (second cycle: days 11-13) + 2 (third cycle: days 14-15) = 15 days. So that's the same answer.Therefore, the answer is the 15th day.Final AnswerThe boxed{15}th day.</think>The magical tree starts with 63 fruits. The number of fruits that fall each day follows an increasing sequence starting from 1, but resets to 1 if there are not enough fruits left. We need to determine the day when all fruits will have fallen.1. Days 1 to 10: The number of fruits falling each day follows the sequence 1, 2, 3, ..., 10. - Total fruits fallen by day 10: (1 + 2 + 3 + ldots + 10 = frac{10 times 11}{2} = 55) - Fruits remaining: (63 - 55 = 8)2. Days 11 to 13: The sequence resets to 1 because there are only 8 fruits left. - Day 11: 1 fruit falls, remaining: (8 - 1 = 7) - Day 12: 2 fruits fall, remaining: (7 - 2 = 5) - Day 13: 3 fruits fall, remaining: (5 - 3 = 2)3. Days 14 to 15: The sequence resets again because there are only 2 fruits left. - Day 14: 1 fruit falls, remaining: (2 - 1 = 1) - Day 15: 1 fruit falls, remaining: (1 - 1 = 0)Thus, all fruits will have fallen by the 15th day.[boxed{15}],<think>Okay, so there's this magical tree with 63 fruits. The problem says that on the first day, 1 fruit falls. Then starting from the second day, each day the number of fruits that fall is 1 more than the previous day. But if the remaining fruits are less than the number that should fall on a certain day, then it starts over by dropping 1 fruit the next day, and continues the original rule. We need to figure out on which day all the fruits will have fallen.Hmm, let's break this down. So initially, there are 63 fruits. On day 1, 1 fruit falls. Then on day 2, 2 fruits should fall. But we need to check if there are enough fruits left each day. If there aren't enough, we reset the count to 1. So this seems like a cycle where the number of fruits that fall increases by 1 each day until there's not enough, then it starts again from 1.Let me try to model this step by step. Let's start with day 1:Day 1: Fruits fallen = 1, Remaining fruits = 63 - 1 = 62Day 2: Should fall 2. Remaining fruits = 62, which is more than 2. So fallen = 2, remaining = 62 - 2 = 60Day 3: Should fall 3. Remaining = 60, which is more than 3. Fallen = 3, remaining = 60 - 3 = 57Continuing this pattern:Day 4: 4 fruits, remaining = 57 - 4 = 53Day 5: 5, remaining = 53 - 5 = 48Day 6: 6, remaining = 48 - 6 = 42Day 7: 7, remaining = 42 - 7 = 35Day 8: 8, remaining = 35 - 8 = 27Day 9: 9, remaining = 27 - 9 = 18Day 10: 10, but remaining is 18. Wait, 18 is more than 10, so 10 falls. Remaining = 18 - 10 = 8Now, next day, day 11: Should be 11, but remaining is 8, which is less than 11. So instead, reset to 1. So day 11: 1 fruit falls, remaining = 8 - 1 = 7Day 12: Now, since we reset, the next day should be 2. Remaining fruits = 7, which is more than 2. So fallen = 2, remaining = 7 - 2 = 5Day 13: Should be 3. Remaining = 5, which is more than 3. So fallen = 3, remaining = 5 - 3 = 2Day 14: Should be 4. Remaining = 2, which is less than 4. So reset to 1. Day 14: fallen = 1, remaining = 2 - 1 = 1Day 15: Next day, should be 2. Remaining = 1, which is less than 2. So reset to 1. Fallen = 1, remaining = 1 - 1 = 0So on day 15, all fruits have fallen. Let me check my steps again to make sure I didn't miscalculate.Starting from day 1 to day 10:Day 1: 1, total fallen 1, remaining 62Day 2: 2, total fallen 3, remaining 60Wait, 63 - 1 - 2 = 60? Yes. Day 3: 3, total fallen 6, remaining 57Day 4: 4, total 10, remaining 53Day 5: 5, total 15, remaining 48Day 6: 6, total 21, remaining 42Day 7: 7, total 28, remaining 35Day 8: 8, total 36, remaining 27Day 9: 9, total 45, remaining 18Day 10: 10, total 55, remaining 8Then day 11: 1, total 56, remaining 7Day 12: 2, total 58, remaining 5Day 13: 3, total 61, remaining 2Day 14: 1, total 62, remaining 1Day 15: 1, total 63, remaining 0Yes, that adds up. Each time the number of fruits to fall is increasing by 1 each day unless there's not enough, then it resets to 1. So after day 10, which was 10 fruits, we had 8 left. Then day 11 starts again at 1. Then day 12 is 2, day 13 is 3, but after day 13, only 2 left. So day 14 starts over at 1, leaving 1. Then day 15 tries to drop 2, but only 1 left, so reset to 1, which takes the last fruit.Wait a second, but in my first calculation, day 15 was able to drop 1, but according to the problem statement: "if the number of fruits on the tree is less than the number that should fall on a certain day, then on that day it will start over by dropping 1 fruit". So on day 15, the number that should fall is 2 (since the previous day was 1, so starting from 1, next day is 2). But remaining fruits is 1, which is less than 2. Therefore, it drops 1 instead. So yes, day 15 drops 1, which is the last fruit.Therefore, the answer is day 15.But let me check again to make sure.Alternatively, perhaps there's a formula for this instead of counting each day. Let me think.This problem is similar to the classic "coins in a fountain" problem or something where you have cycles of increasing numbers. Each cycle, you subtract the sum of 1 to n until you can't, then start over.But here, each time the number of fruits to fall is 1 more than the previous day. So each "cycle" is a sequence of days where the fruits fallen are 1, 2, 3, ..., k until the remaining fruits are less than k+1. Then the next day, it starts again at 1.So the total number of days would be the sum of the lengths of these cycles until the total fruits equal 63.Alternatively, maybe we can model it as:First cycle: try to drop 1, 2, 3, ..., k until the sum exceeds 63. But in our case, it's not exactly the sum, because after each cycle where the remaining fruits can't support the next number, it starts over. Wait, actually, in our problem, if on day n, the number to drop is m, but remaining fruits are less than m, then we drop 1 instead, and the next day, we try m=2 again? Wait no, according to the problem statement: "if the number of fruits on the tree is less than the number that should fall on a certain day, then on that day it will start over by dropping 1 fruit and continue according to the original rule, and so on."So when it resets, the next day it starts at 1 again, and then increments each day. So each time you have a reset, you start a new cycle.Therefore, the process is: keep trying to increment the number of fruits to drop each day, starting at 1, until you can't, then reset to 1 and repeat.Therefore, the days can be broken into sequences where each sequence is 1, 2, 3, ..., k where the sum of 1+2+...+k is less than or equal to the remaining fruits. Wait, no, not exactly, because each day you subtract the number, not the sum.Wait, perhaps not. Let's clarify. Each day, you attempt to drop m fruits, where m starts at 1 and increases by 1 each day. If you can't drop m fruits (because remaining < m), then you drop 1 instead, and the next day you start m at 2? Wait, no. The problem says: "if the number of fruits on the tree is less than the number that should fall on a certain day, then on that day it will start over by dropping 1 fruit and continue according to the original rule, and so on."So if on day n, the number that should fall is m (which is 1 more than the previous day), but remaining fruits are less than m, then on that day, drop 1, and then the next day, start over by incrementing from 1 again. So the reset happens on the day when there aren't enough fruits, and you drop 1 instead, and then the following day, you go back to the original rule of increasing by 1 each day.Wait, let's parse the problem statement again:"Starting from the second day, the number of fruits that fall each day is 1 more than the previous day. However, if the number of fruits on the tree is less than the number that should fall on a certain day, then on that day it will start over by dropping 1 fruit and continue according to the original rule, and so on."So "on that day it will start over by dropping 1 fruit". So if day n is supposed to drop m fruits, but there are less than m, then on day n, it drops 1 instead, and then on day n+1, it will drop 2 (since it's 1 more than the previous day, which was 1). So the cycle restarts with 1 on the day when it can't drop the required number.So in effect, each time you can't drop the required number, you drop 1 and then start a new cycle.Therefore, the process is: you have cycles. Each cycle is a sequence of days where the number of fruits dropped increases by 1 each day, starting from 1, until you can't drop the next number, then you drop 1 and start a new cycle.So the first cycle is days 1 to k, where the fruits dropped are 1, 2, 3, ..., k, and the total dropped in this cycle is 1+2+...+k. But wait, actually, not exactly, because if on day k, you drop k fruits, and then on day k+1, you attempt to drop k+1, but if you can't, you drop 1 instead, starting a new cycle.Wait, but in the problem statement, the first cycle is 1, then 2, etc., but the total fruits are 63. So in the example we did manually, the first cycle was 1, 2, 3, ..., 10 (days 1-10), total dropped 55 fruits. Then the remaining 8. Then the next cycle is days 11-13: 1, 2, 3 (total 6), remaining 2. Then next cycle days 14-15: 1, 1 (total 2). So total days 15.But perhaps there is a formula here.Alternatively, maybe we can model the total number of days as the sum of multiple arithmetic sequences until the total sum reaches 63.Wait, but each time we have a "cycle" that is an arithmetic sequence starting at 1, increasing by 1 each day, until the next term would exceed the remaining fruits. Then the remaining fruits are subtracted by 1, and a new cycle starts.So, for example, first cycle: 1+2+3+...+k ≤ 63. Then remaining fruits: 63 - sum(1..k). Then next cycle: 1+2+3+...+m ≤ remaining fruits. And so on.But in reality, it's not exactly the sum of the cycles, because after each cycle, the remaining fruits are decreased by the sum of that cycle. However, in our problem, each cycle is interrupted when the next term can't be subtracted. So it's like partial sums.Wait, maybe not. Let's think again.First, let's do it step by step for the given problem to confirm.Total fruits: 63First cycle: day 1: 1, remaining 62day 2: 2, remaining 60day 3: 3, remaining 57day 4: 4, remaining 53day 5: 5, remaining 48day 6: 6, remaining 42day 7: 7, remaining 35day 8: 8, remaining 27day 9: 9, remaining 18day 10:10, remaining 8So the first cycle is 10 days, sum 55, remaining 8.Now, next cycle:day 11:1, remaining 7day12:2, remaining5day13:3, remaining2Sum here is 1+2+3=6, remaining 2.Next cycle:day14:1, remaining1day15:2, but only 1 left, so drop1, remaining0.So total days 15.Alternatively, if we model each cycle as sum(1..k) where sum(1..k) ≤ remaining fruits. Then total days would be k each cycle, but in reality, after the first cycle, the remaining fruits are 63 - sum(1..10)=63-55=8.Then next cycle: find the largest m such that sum(1..m) ≤8. sum(1..4)=10>8, sum(1..3)=6≤8. So m=3, days 11-13 (3 days), remaining 8-6=2.Next cycle: find largest n such that sum(1..n)≤2. sum(1..2)=3>2, sum(1..1)=1≤2. So n=1, days14, remaining2-1=1.Next cycle: sum(1..1)=1≤1. So day15, remaining1-1=0.Total days:10+3+1+1=15.Ah, so this seems like a pattern. Each cycle, we take the largest possible sum of consecutive integers starting at 1 that is less than or equal to the remaining fruits, then subtract that sum and proceed. The number of days in each cycle is the number of terms in that sum. Then we add up all those days.But in the problem, the process is slightly different. Because in the problem, the reset occurs when the next term is too big, not the sum. Wait, no. Wait, the problem says each day you try to drop 1 more than the previous day. If you can't drop that number, you drop 1 and start over. So in effect, each cycle is terminated when the next number to drop would exceed the remaining fruits. Then on that day, you drop 1 instead, and start a new cycle.Therefore, each cycle's length is determined by the maximum number of days you can go increasing by 1 each day until the next day's number would exceed the remaining fruits.But the way the problem is structured, when you reset, you drop 1 on the same day when you can't drop the required number, and then start a new cycle the next day. So each time you hit a day where you can't drop the required number, you substitute 1 for that day, and the next day you start again at 2? Wait, no.Wait, let's parse the problem statement again:"Starting from the second day, the number of fruits that fall each day is 1 more than the previous day. However, if the number of fruits on the tree is less than the number that should fall on a certain day, then on that day it will start over by dropping 1 fruit and continue according to the original rule, and so on."So if on day n, the number to fall is m (which is 1 more than day n-1), but remaining fruits are less than m, then on day n, instead of m, you drop 1, and then on day n+1, you start over by dropping 1 more than day n, which would be 2. Wait, no, because the original rule is starting from the second day, each day is 1 more than the previous. But if you reset, then on day n, you drop 1, so day n+1 should be 1 more than day n, which is 2.Wait, but according to the original rule, starting from the second day, each day is 1 more than the previous. So if day n-1 was k, then day n is k+1. But if on day n, you can't drop k+1, you drop 1 instead, then day n+1 would be 1 more than day n's drop. Since day n's drop was 1, day n+1 would be 2. So effectively, after a reset, the next day starts a new cycle with 1, 2, 3, etc.Therefore, the process is that each time you reset, you start a new cycle beginning with 1, then 2, etc., regardless of where you were before. Therefore, each cycle is a sequence of days where you drop 1, 2, 3, ..., m, where m is the maximum such that 1+2+...+m <= remaining fruits. Wait, but actually, it's not the sum, because each day you subtract the number, not the sum. Wait, but the cycle is of consecutive days with increasing numbers. However, the total fruits removed in a cycle is the sum of 1+2+...+m.But in reality, in the problem, the fruits are subtracted each day, so the remaining fruits decrease each day by the number that falls. So the cycles are sequences of days where each day the number fallen increases by 1, until the next number would exceed the remaining fruits. Then you drop 1 on that day, and start a new cycle.But in the example:First cycle: 1,2,3,4,5,6,7,8,9,10 (10 days). Sum is 55. Remaining fruits 8.Then next cycle: starting with 1, but since the remaining is 8, how many days can we go? Let's see:Day11:1, remaining7Day12:2, remaining5Day13:3, remaining2Now, next day, day14: should be 4, but remaining is 2, so drop1, remaining1.Then day15: should be 2 (1 more than previous day's 1), but remaining1 <2, so drop1, remaining0.Wait, but in the manual count, day14 was 1, and day15 was1. But according to the rule, after dropping1 on day14 (because remaining was2 <4), then day15 should be 1 more than day14's drop, which was1, so day15 should drop2. But remaining is1, which is less than2, so day15 drops1. Then day16 would start over.But in our manual calculation, it ended at day15. Therefore, the formula approach might not directly apply here because each time you reset, you drop1, and then the next day you attempt to drop2, but if you can't, you reset again.Therefore, the process is that after a reset day (where you drop1 instead of the required number), the next day you attempt to drop2, regardless of previous cycles.Therefore, each cycle is either:1. A sequence of days where the number dropped increases by1 each day, starting from1, until the next day's number would exceed the remaining fruits.OR2. If you can't even drop1, but the problem states that on the day when you can't drop the required number, you drop1 instead. So even if there is 1 fruit left, you drop1. Wait, but if there are zero fruits left, then obviously it's done.But in the problem statement, it says "on that day it will start over by dropping1 fruit", so even if there is 1 fruit left, you can drop1. Only when there are zero fruits left does it stop.Therefore, in our manual calculation:After day13: remaining2Day14: supposed to drop4 (since previous day was3), but remaining2 <4, so drop1, remaining1Day15: supposed to drop2 (1 more than previous day's1), but remaining1 <2, so drop1, remaining0So day15 ends it.Therefore, the cycles aren't complete arithmetic sequences, but rather partial sequences where each cycle ends when the next number is too big, and you drop1 instead. Then the next cycle starts with attempting to drop2, but if that's too big, you drop1 again, etc.Therefore, perhaps the number of days can be calculated by considering how many complete increasing sequences we can have before the remaining fruits get low enough that we have multiple resets.Alternatively, perhaps there's a mathematical way to model this.Let me consider the process as follows:Each time we have a "run" of days where the number dropped increases by1 each day, starting from1, until the next number is too big, then we have a reset day where we drop1. Then we start a new run.But each run's length is the maximum m such that 1+2+...+m <= remaining fruits. Wait, no, because in the problem, the fruits are subtracted each day, so the remaining fruits decrease each day by the number dropped. Therefore, the first run subtracts1, then2, etc., so the remaining fruits after each run is initial fruits minus the sum of the run.But in the problem, the runs can be interrupted not by the sum, but by the next term being too big. Wait, actually, it's exactly that.For example, if we have remaining fruits R, then the maximum number m we can drop next is the largest integer such that m <= R. However, the next day, we need to drop m+1. So the run continues as long as the next number is <= remaining fruits.Wait, perhaps another approach.Let’s model each "run" as a sequence of days where the number of fruits dropped each day is increasing by1, starting from1, and the run ends when the next number to be dropped would exceed the remaining fruits. Then on the day when the next number is too big, we drop1 instead, and start a new run.Therefore, each run has length k, where k is the largest integer such that k <= remaining fruits after the previous runs. Wait, no. Let me think.Suppose we have R remaining fruits. The first day of the run, we drop1. Then next day, drop2, etc., until we can't drop k+1. So the run length is k, where 1+2+...+k <= R, but 1+2+...+k+(k+1) > R.But no, because the remaining fruits are decreasing each day by the amount dropped. So it's not the sum, but the sequence of subtractions.Wait, let's take the first run:Start with R=63Day1: drop1, R=62Day2: drop2, R=60Day3: drop3, R=57...Each day, R decreases by the number dropped, which is incrementing by1 each day.The run continues until the next day's required drop is greater than the current R.So in the first run:After day10, R=8. Next day's drop would be11, but R=8 <11. So run ends at day10, length10.Then, on day11, drop1, R=7.Next run:Day11:1, R=7Day12:2, R=5Day13:3, R=2Next day would be4, but R=2 <4, so run ends at day13, length3.Then day14: drop1, R=1Next run:Day14:1, R=1Next day would be2, but R=1 <2, so run ends at day14, length1Day15: drop1, R=0So total days:10+3+1+1=15.Wait, but the runs are of lengths10,3,1, and the resets are days11,14,15. Hmm, but day15 is the last day. So the formula is sum of the run lengths plus the number of resets? Not exactly.Alternatively, each run is followed by a reset day. Except the last reset day which may end the process.Wait, no. Each time a run ends, the next day is a reset day (dropping1). So the number of runs is equal to the number of resets plus1?Wait, let's see:First run: days1-10 (10 days), then day11 is a reset (drop1).Second run: days11-13 (drop1,2,3), then day14 is a reset (drop1).Third run: day14 (drop1), then day15 is a reset (drop1).Wait, day14 is part of the second reset? This is getting confusing.Alternatively, each time a run cannot continue, we have a reset day, which is part of the next run.Wait, perhaps it's better to model the entire process as a sequence of days where each day's drop is either the previous day's drop +1 or reset to1.But given the problem statement, after a reset, the next day's drop is1 more than the reset day's drop (which was1), so2.Wait, according to the problem: "Starting from the second day, the number of fruits that fall each day is 1 more than the previous day. However, if the number of fruits on the tree is less than the number that should fall on a certain day, then on that day it will start over by dropping1 fruit and continue according to the original rule, and so on."So the "original rule" is that each day is1 more than the previous. So if you have to reset on day n, then day n's drop is1, and day n+1's drop is1 more than day n's, which is2.Therefore, after a reset day (dropping1), the next day you start a new incrementing sequence. So the reset day is part of the sequence.Wait, for example:Days1-10:1,2,3,4,5,6,7,8,9,10.Then day11:1 (reset), day12:2, day13:3.Then day14:1 (reset), day15:2 (but only1 left, so reset again to1).Wait, but day15: supposed to drop2, but can't, so reset to1.But according to the rule, after day14 (which was a reset to1), day15 should be1 more than day14, which is2. But since remaining fruits=1 <2, day15 drops1 instead.Therefore, day15 is another reset day. So total days:First sequence:1-10 (10 days)Second sequence:11-13 (3 days:1,2,3)Third sequence:14-14 (1 day:1)Fourth sequence:15-15 (1 day:1)Wait, but sequences are supposed to be incrementing each day unless reset. So after day14 (drop1), day15 should be2. But since only1 left, drop1. Then day16 would be2, but there are0 left.But since we've already dropped all fruits on day15, the process stops at day15.Therefore, the total days are15.But this is getting complex. Maybe a better way is to simulate it step by step as done initially.Alternatively, perhaps there's a mathematical approach here.Notice that each time we have a run of k days dropping1,2,...,k fruits, the total fruits dropped are k(k+1)/2. Then, if after that run, the remaining fruits are S = 63 - k(k+1)/2. Then, the next run would attempt to drop1,2,...,m where m(m+1)/2 <= S. But in reality, the process is different because after each run, the next day starts at1 again.Wait, but in our problem, after a run ends (when the next number is too big), we drop1 and start a new run. So the total fruits dropped in a run of length k is1+2+...+k =k(k+1)/2, but when a run is interrupted, we only drop1 instead of k+1, so the total fruits in that interrupted run is1+2+...+k, and then the reset day drops1.Wait, but in our manual calculation:First run:1-10, sum55, then reset day11 drops1.Second run:11-13 (1,2,3), sum6, then reset day14 drops1.Third run:14-14 (1), sum1, then reset day15 drops1.Total fruits:55+1+6+1+1=64. Wait, but we only had63 fruits. There's a miscalculation here.Wait, no. In reality:After the first run (days1-10): total dropped55, remaining8.Then day11: drop1, remaining7.Then days12-14: drop2,3,1. Wait, day12:2, day13:3, day14:1. Wait, this doesn't make sense.Wait, no. Let's re-express:First run: days1-10: drop1,2,...,10 (total55), remaining8.Then day11: can't drop11, so drop1, remaining7.Then day12: previous day was1, so drop2, remaining5.Day13: drop3, remaining2.Day14: supposed to drop4, but remaining2, so drop1, remaining1.Day15: previous day was1, so drop2, but remaining1, so drop1, remaining0.Therefore, total dropped:55+1+2+3+1+1=63.Therefore, the days are:First run:10 days (sum55)Then days11 (1),12 (2),13 (3): sum1+2+3=6Then days14 (1),15 (1): sum1+1=2Total days:10+3+2=15.But the total sum is55+6+2=63.But how does this break down into runs? The first run is10 days. Then after reset, the next run is3 days (days11-13:1,2,3), then another reset on day14, which starts a new run but can only do1 day (day14:1), reset on day15, which does another1 day (day15:1).But in terms of runs:First run:1-10 (10 days)Second run:11-13 (3 days)Third run:14-14 (1 day)Fourth run:15-15 (1 day)But each run after a reset starts at1 and increments each day. However, the third run (day14) starts at1, but the next day (15) was supposed to be2, but had to reset. Therefore, day15 is part of the fourth run.This seems complicated, but the total days are indeed15.Is there a way to generalize this?Let me consider that each time we have a complete run of k days, dropping1+2+...+k fruits, then we have a reset day dropping1, then the next run is trying to drop1,2,...,m, etc.But since after each reset, the remaining fruits are getting smaller, maybe we can represent this recursively.Let’s define a function f(n) which gives the number of days required to drop n fruits.If n ==0, return0.Otherwise, find the largest k such that k(k+1)/2 <=n.Then, the days would be k days for this run, plus1 day for the reset (dropping1 fruit), plus f(n - (k(k+1)/2 +1)).Wait, but in reality, after the run of k days, you have a reset day where you drop1, so total fruits dropped are k(k+1)/2 +1, and remaining fruits are n - (k(k+1)/2 +1). Then, the days would be k +1 + f(n - (k(k+1)/2 +1)).But let's test this with n=63.First, find largest k where k(k+1)/2 <=63.k=10, since10*11/2=55<=63, and11*12/2=66>63.So first run:10 days, dropped55, remaining63-55=8.Then reset day:1 fruit, remaining8-1=7.Days so far:10+1=11.Now, recurse on7.Find largest k where k(k+1)/2 <=7.k=3, since3*4/2=6<=7, 4*5/2=10>7.So run of3 days, dropping1+2+3=6, remaining7-6=1.Days so far:11+3=14.Reset day:1 fruit, remaining1-1=0.Days total:14+1=15.So total days15. That matches.Wait, this seems to work.So the formula is:f(n) = k +1 + f(n - (k(k+1)/2 +1)), where k is the largest integer with k(k+1)/2 <=n.Base case: f(0)=0.Let’s verify with n=1.f(1): k=1 since1*2/2=1<=1.Then days=1+1 +f(1 - (1 +1))=1+1 +f(-1). But since we can't have negative fruits, f(-1)=0.So total days=2. But in reality, for n=1, day1 drops1, done. So f(1)=1. Hmm, discrepancy here.Wait, maybe the formula needs adjustment.Wait, if we follow the formula for n=1:k is the largest integer where k(k+1)/2 <=1.k=1.Then f(1)=1 +1 +f(1 - (1 +1))=2 +f(-1)=2 +0=2. But in reality, it's1 day.So the formula overcounts. The issue is that after the first run of k days, we add a reset day, but if the remaining fruits after the run are zero, we shouldn't add the reset day.Wait, let's redefine the formula more accurately.Define f(n):If n ==0: return0Else:Find the largest k such that sum_{i=1}^k i <=nLet S = sum_{i=1}^k i =k(k+1)/2If S ==n:Then days =kElse:Days =k +1 (the reset day) + f(n - S -1)So in the case where the sum is exactly n, you don't need a reset day. Otherwise, you have the run of k days, a reset day, and then recurse on n - S -1.Testing this:For n=1:Largest k with sum <=1 isk=1, sum=1==n, so days=k=1. Correct.For n=2:Largest k with sum <=2 isk=1 (sum=1<=2). Since sum=1 <2, days=1 +1 +f(2 -1 -1)=2 +f(0)=2+0=2. Which is correct: day1:1, day2:1.Wait, but according to the problem rules, for n=2:Day1:1, remaining1Day2:2, but remaining1 <2, so reset to1, remaining0. So days=2. Which matches.For n=3:Largest k with sum <=3 isk=2 (sum=3). S==n, so days=2. Correct:Day1:1, day2:2, total3.For n=4:Largest k with sum <=4 isk=2 (sum=3 <=4). Since3 <4, days=2 +1 +f(4 -3 -1)=3 +f(0)=3.Which corresponds:Day1:1, day2:2 (sum3), reset day3:1, total4. Correct.Yes, day1:1, day2:2, day3:1. Total4.Similarly, n=5:Largest k=2 (sum3). days=2 +1 +f(5-3-1)=3 +f(1)=3 +1=4.Which is:Day1:1, day2:2, day3:1, day4:1. Sum1+2+1+1=5.But according to problem rules:Day1:1, remaining4Day2:2, remaining2Day3:3, but remaining2 <3, reset to1, remaining1Day4:2, but remaining1 <2, reset to1, remaining0. So days4. Correct.But according to the formula, it's also4. Wait, but according to the problem rules, the days would be:Day1:1, day2:2, day3:1, day4:1.But according to the formula, it's k=2, then reset day3, then f(1)=1. So total2+1+1=4.Yes, matches.Therefore, the formula works.Let's test it on our original problem n=63.First iteration:Find largest k wherek(k+1)/2 <=63.k=10, sum=55<=63.Since55 <63, days=10 +1 +f(63 -55 -1)=11 +f(7).Now compute f(7):Largest k wherek(k+1)/2 <=7.k=3, sum=6 <=7.6 <7, so days=3 +1 +f(7 -6 -1)=4 +f(0)=4 +0=4.Thus total days so far11 +4=15.Next f(0)=0.Thus total days=15. Which matches our manual calculation.Therefore, the formula works.Therefore, the answer is15.Final AnswerThe day on which all fruits will have fallen is boxed{15}.</think>The problem involves a magical tree with 63 fruits. The fruits fall according to a specific rule: 1 fruit falls on the first day, and each subsequent day, the number of fruits that fall is 1 more than the previous day. If the remaining fruits are less than the number that should fall on a certain day, the count resets to 1, and the process continues. We need to determine the day on which all fruits will have fallen.1. Initial Analysis: - The first cycle of fruits falling starts with 1 fruit and increases by 1 each day until the number of fruits left is insufficient. - If the remaining fruits are less than the number required to fall, the count resets to 1, and the cycle continues.2. Manual Calculation: - Days 1-10: The number of fruits falling increases by 1 each day, totaling 55 fruits. Remaining fruits = 63 - 55 = 8. - Days 11-13: Reset to 1, then 2, 3. Total fallen = 1 + 2 + 3 = 6. Remaining fruits = 8 - 6 = 2. - Days 14-15: Reset to 1, then 1 again. Total fallen = 1 + 1 = 2. Remaining fruits = 2 - 2 = 0.3. General Formula Approach: - Define a function ( f(n) ) to calculate the number of days required for ( n ) fruits. - For each cycle, find the largest ( k ) such that ( sum_{i=1}^k i leq n ). - If the sum equals ( n ), the days are ( k ). Otherwise, add the reset day and recurse on the remaining fruits.4. Applying the Formula: - For ( n = 63 ): - First cycle: ( k = 10 ) (sum = 55), remaining fruits = 8. Days = 10 + 1 + f(7). - Second cycle: ( k = 3 ) (sum = 6), remaining fruits = 1. Days = 3 + 1 + f(0). - Third cycle: ( f(1) = 1 ).5. Total Days Calculation: - Total days = 10 (first cycle) + 1 (reset) + 3 (second cycle) + 1 (reset) + 1 (final day) = 15.Thus, the day on which all fruits will have fallen is (boxed{15}).]

🔑:【Solution】Solution: The total number of fruits that fall in the first ten days is:1+2+3+4+5+6+7+8+9+10=55 text { (fruits) }There are still 63-55=8 (fruits) remaining.Then it starts over, with 1 fruit on the first day, 2 fruits on the second day, and 3 fruits on the third day,1+2+3=6,There are still 8-6=2 remaining, which is less than the number of days, so it starts over,Then 1 fruit on the first day, and then it starts over again, 1 fruit on the first day,So: 10+3+2=15 (days)The answer is: 15.

🧠:[<think>Okay, so I need to find the derivative of y = (x² + 2)/(2√(1 - x⁴)). Alright, let's see. Hmm, derivatives. Right, this is a quotient, so I should probably use the quotient rule. But maybe I can simplify it first or use the chain rule. Let me recall the quotient rule: if I have a function f(x)/g(x), the derivative is (f’(x)g(x) - f(x)g’(x))/[g(x)]². Yeah, that seems right.First, let me identify f(x) and g(x) here. The numerator is x² + 2, so f(x) = x² + 2. The denominator is 2√(1 - x⁴). So g(x) = 2√(1 - x⁴). Okay, so I need to compute f’(x) and g’(x). Let's start with f’(x). The derivative of x² is 2x, and the derivative of 2 is 0, so f’(x) = 2x. That's straightforward.Now, g(x) = 2√(1 - x⁴). To find g’(x), I need to apply the chain rule here. The outer function is 2 times the square root of something, and the inner function is 1 - x⁴. Let me write it as 2*(1 - x⁴)^(1/2). So, the derivative of that would be 2 * (1/2)(1 - x⁴)^(-1/2) * derivative of the inner function. The 2 and 1/2 cancel out, so we have (1 - x⁴)^(-1/2) * (-4x³). So, g’(x) = -4x³ / √(1 - x⁴). Wait, but hold on, the original g(x) is 2√(1 - x⁴), so when I took the derivative, the 2 was multiplied by the derivative of the square root. Let me check that again.Yes, so g(x) = 2*(1 - x⁴)^(1/2). So derivative is 2*(1/2)(1 - x⁴)^(-1/2)*(-4x³). That simplifies to 2*(1/2) which is 1, then multiplied by (-4x³)/(1 - x⁴)^(1/2). So g’(x) = (-4x³)/√(1 - x⁴). That seems right. Okay.So now, applying the quotient rule: y’ = [f’(x)g(x) - f(x)g’(x)] / [g(x)]².Let's compute each part step by step.First, f’(x)g(x) is 2x multiplied by 2√(1 - x⁴). So that's 2x * 2√(1 - x⁴) = 4x√(1 - x⁴).Next, f(x)g’(x) is (x² + 2) multiplied by (-4x³)/√(1 - x⁴). So that's (x² + 2)*(-4x³)/√(1 - x⁴).Putting these into the numerator: 4x√(1 - x⁴) - (x² + 2)*(-4x³)/√(1 - x⁴). Wait, but the second term is subtracted, so it's minus f(x)g’(x). But f(x)g’(x) is negative, so subtracting a negative becomes a positive. Let me clarify.The quotient rule is [f’ g - f g’] / g². Since g’ is negative, the term -f g’ would be positive. Let me make sure.Yes, so numerator is:f’(x)g(x) - f(x)g’(x) = (2x)(2√(1 - x⁴)) - (x² + 2)(-4x³/√(1 - x⁴))So that's 4x√(1 - x⁴) + (x² + 2)(4x³)/√(1 - x⁴)Hmm, now the numerator has two terms: one with √(1 - x⁴) and another with 1/√(1 - x⁴). To combine them, I might need a common denominator. Let's see.First term: 4x√(1 - x⁴) can be written as 4x(1 - x⁴)^(1/2)Second term: (x² + 2)(4x³)/(1 - x⁴)^(1/2)So, to combine them, the first term would need to be multiplied by (1 - x⁴)/(1 - x⁴) to have the same denominator. So:4x(1 - x⁴)^(1/2) = 4x(1 - x⁴)^(1/2) * [ (1 - x⁴)/(1 - x⁴) ) ]^(1/2) ??? Wait, maybe another approach.Wait, actually, 4x√(1 - x⁴) can be written as 4x(1 - x⁴)/√(1 - x⁴). Because √(1 - x⁴) times √(1 - x⁴) is (1 - x⁴). So 4x√(1 - x⁴) = 4x(1 - x⁴)/√(1 - x⁴). Then, both terms in the numerator would have denominator √(1 - x⁴), so we can combine them.So:4x(1 - x⁴)/√(1 - x⁴) + 4x³(x² + 2)/√(1 - x⁴)Which is [4x(1 - x⁴) + 4x³(x² + 2)] / √(1 - x⁴)Factor out 4x from the numerator:4x[ (1 - x⁴) + x²(x² + 2) ] / √(1 - x⁴)Now, let's compute the expression inside the brackets:(1 - x⁴) + x²(x² + 2) = 1 - x⁴ + x⁴ + 2x² = 1 + 2x²Because -x⁴ + x⁴ cancels out, leaving 1 + 2x².So, the numerator becomes 4x(1 + 2x²) / √(1 - x⁴)Therefore, the entire derivative y’ is [4x(1 + 2x²) / √(1 - x⁴)] divided by [g(x)]².Wait, no. Wait, the quotient rule denominator is [g(x)]². So the entire expression is:[4x(1 + 2x²) / √(1 - x⁴)] / [ (2√(1 - x⁴))² ]Let me compute [g(x)]². g(x) = 2√(1 - x⁴), so squared is 4(1 - x⁴).Therefore, denominator is 4(1 - x⁴).So putting it all together:y’ = [4x(1 + 2x²) / √(1 - x⁴)] / [4(1 - x⁴)]Simplify the fractions. The 4 in numerator and denominator cancels out:= [x(1 + 2x²) / √(1 - x⁴)] / (1 - x⁴)Which is equal to x(1 + 2x²) / [ √(1 - x⁴) * (1 - x⁴) ]Note that √(1 - x⁴) * (1 - x⁴) is (1 - x⁴)^(1/2) * (1 - x⁴)^1 = (1 - x⁴)^(3/2)Therefore, y’ = x(1 + 2x²) / (1 - x⁴)^(3/2)Alternatively, we can write this as x(1 + 2x²) / (1 - x⁴)^(3/2), which might be the simplified form.Let me check if that's correct. Let's verify each step.First, f’(x) = 2x, correct.g(x) = 2√(1 - x⁴), so g’(x) = 2*(1/2)(1 - x⁴)^(-1/2)*(-4x³) = (-4x³)/√(1 - x⁴), correct.Then, the numerator in the quotient rule:2x*2√(1 - x⁴) - (x² + 2)*(-4x³)/√(1 - x⁴) = 4x√(1 - x⁴) + 4x³(x² + 2)/√(1 - x⁴)Then, combining the terms by expressing 4x√(1 - x⁴) as 4x(1 - x⁴)/√(1 - x⁴):So numerator is [4x(1 - x⁴) + 4x³(x² + 2)] / √(1 - x⁴)Factor 4x:4x[ (1 - x⁴) + x³(x² + 2)/x ] Wait, but x³(x² + 2) is x^5 + 2x³. Wait, but when factoring 4x, inside the brackets:Wait, in the numerator after factoring 4x:Original expression after factoring 4x:4x[ (1 - x⁴) + x²(x² + 2) ] / √(1 - x⁴)Wait, x²(x² + 2) = x⁴ + 2x². Then:(1 - x⁴) + x⁴ + 2x² = 1 + 2x², correct. So numerator becomes 4x(1 + 2x²)/√(1 - x⁴). Then denominator of quotient rule is [2√(1 - x⁴)]² = 4(1 - x⁴). So dividing numerator by denominator:[4x(1 + 2x²)/√(1 - x⁴)] / [4(1 - x⁴)] = x(1 + 2x²)/[ (1 - x⁴)√(1 - x⁴) ] = x(1 + 2x²)/(1 - x⁴)^(3/2)Yes, that seems correct. Let me check if I can simplify further or if there's another way.Alternatively, maybe we could have used logarithmic differentiation or another method, but the quotient rule seems straightforward here. Let me see if the answer can be written differently, but I think x(1 + 2x²)/(1 - x⁴)^(3/2) is the simplified form.Wait, let's verify with another approach. Maybe rewrite the original function and see.Original function: y = (x² + 2)/(2√(1 - x⁴))Alternatively, we can write this as y = (x² + 2)/[2(1 - x⁴)^(1/2)]Then, using the quotient rule or product rule. Let me try using the product rule by rewriting as y = (x² + 2) * [2(1 - x⁴)^(1/2)]^(-1)So y = (x² + 2) * (1/2)(1 - x⁴)^(-1/2)Then, derivative is:d/dx [x² + 2] * (1/2)(1 - x⁴)^(-1/2) + (x² + 2) * d/dx [ (1/2)(1 - x⁴)^(-1/2) ]First term: 2x * (1/2)(1 - x⁴)^(-1/2) = x(1 - x⁴)^(-1/2)Second term: (x² + 2) * [ (1/2)*(-1/2)(1 - x⁴)^(-3/2)*(-4x³) ]Let's compute the derivative inside:d/dx [ (1/2)(1 - x⁴)^(-1/2) ] = (1/2)*(-1/2)(1 - x⁴)^(-3/2)*(-4x³) = (1/2)*(2x³)/(1 - x⁴)^(3/2) = x³/(1 - x⁴)^(3/2)Wait, let's compute step by step:First, derivative of (1 - x⁴)^(-1/2) is (-1/2)(1 - x⁴)^(-3/2)*(-4x³) = (2x³)/(1 - x⁴)^(3/2). Then multiplied by (1/2):So derivative is (1/2)*(2x³)/(1 - x⁴)^(3/2) = x³/(1 - x⁴)^(3/2)Therefore, the second term is (x² + 2) * x³/(1 - x⁴)^(3/2)Therefore, total derivative y’ = x/(1 - x⁴)^(1/2) + (x³(x² + 2))/(1 - x⁴)^(3/2)Now, to combine these terms, we can factor out x/(1 - x⁴)^(3/2)First term: x/(1 - x⁴)^(1/2) = x(1 - x⁴)/(1 - x⁴)^(3/2)Second term: x³(x² + 2)/(1 - x⁴)^(3/2)So combining them:[ x(1 - x⁴) + x³(x² + 2) ] / (1 - x⁴)^(3/2)Simplify numerator:x(1 - x⁴) + x³(x² + 2) = x - x^5 + x^5 + 2x³ = x + 2x³So numerator is x + 2x³ = x(1 + 2x²)Thus, derivative is x(1 + 2x²)/(1 - x⁴)^(3/2), same as before. So that confirms the answer.Therefore, the derivative is y’ = x(1 + 2x²)/(1 - x⁴)^(3/2)Just to check if there are any algebraic simplifications. Let's see: 1 + 2x² and 1 - x⁴. Notice that 1 - x⁴ factors as (1 - x²)(1 + x²) = (1 - x)(1 + x)(1 + x²). But in the numerator we have 1 + 2x², which doesn't factor into any of those terms, so probably the answer is as simplified as it gets.So, the final answer should be boxed as:boxed{dfrac{x(1 + 2x^{2})}{(1 - x^{4})^{3/2}}}Final Answerboxed{dfrac{x(1 + 2x^{2})}{(1 - x^{4})^{3/2}}}</think>To find the derivative of ( y = frac{x^{2} + 2}{2 sqrt{1 - x^{4}}} ), we use the quotient rule. Given:- ( f(x) = x^2 + 2 )- ( g(x) = 2sqrt{1 - x^4} )First, compute the derivatives of ( f(x) ) and ( g(x) ):- ( f'(x) = 2x )- ( g'(x) ) is found using the chain rule. Rewriting ( g(x) ) as ( 2(1 - x^4)^{1/2} ), we get: [ g'(x) = 2 cdot frac{1}{2}(1 - x^4)^{-1/2} cdot (-4x^3) = frac{-4x^3}{sqrt{1 - x^4}} ]Applying the quotient rule:[y' = frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}]Substitute ( f(x) ), ( f'(x) ), ( g(x) ), and ( g'(x) ):[y' = frac{(2x cdot 2sqrt{1 - x^4}) - (x^2 + 2) cdot left( frac{-4x^3}{sqrt{1 - x^4}} right)}{(2sqrt{1 - x^4})^2}]Simplify the numerator and denominator:- Numerator: [ 4xsqrt{1 - x^4} + frac{4x^3(x^2 + 2)}{sqrt{1 - x^4}} ] Combine terms by expressing ( 4xsqrt{1 - x^4} ) as ( frac{4x(1 - x^4)}{sqrt{1 - x^4}} ): [ frac{4x(1 - x^4) + 4x^3(x^2 + 2)}{sqrt{1 - x^4}} = frac{4x(1 + 2x^2)}{sqrt{1 - x^4}} ]- Denominator: [ (2sqrt{1 - x^4})^2 = 4(1 - x^4) ]Combine and simplify:[y' = frac{frac{4x(1 + 2x^2)}{sqrt{1 - x^4}}}{4(1 - x^4)} = frac{x(1 + 2x^2)}{(1 - x^4)^{3/2}}]Thus, the derivative is:[boxed{dfrac{x(1 + 2x^{2})}{(1 - x^{4})^{3/2}}}],<think>Okay, so I need to find the derivative of the function y equals (x squared plus 2) divided by (2 times the square root of (1 minus x to the fourth power)). Hmm, let's see. I remember that when you have a function that's a quotient of two expressions, you can use the quotient rule. Alternatively, maybe I can simplify it first before differentiating. Let me think.First, let me write down the function clearly: y = (x² + 2) / [2√(1 - x⁴)]. Hmm. So, the numerator is (x² + 2) and the denominator is 2 times the square root of (1 - x⁴). Maybe it's easier if I rewrite the denominator as 2 times (1 - x⁴) raised to the 1/2 power. That might make applying the rules easier.So, y = (x² + 2) / [2(1 - x⁴)^(1/2)]. Now, since the denominator is 2 times something, maybe I can factor out the 2 and write it as (1/2) times (x² + 2) divided by (1 - x⁴)^(1/2). Then, maybe I can use the quotient rule on (x² + 2) over (1 - x⁴)^(1/2) and then multiply by 1/2 afterwards. But the quotient rule can get messy. Alternatively, maybe I can use the product rule by considering the numerator times the denominator raised to a negative exponent.Let me try that approach. So, if I rewrite y as (1/2)(x² + 2)(1 - x⁴)^(-1/2), then I can use the product rule. The product rule states that the derivative of u*v is u'v + uv'. So, here u would be (x² + 2) and v would be (1 - x⁴)^(-1/2). Then, the derivative y' would be (1/2)[u'v + uv']. Let's compute each part.First, find u'. u is (x² + 2), so u' is 2x. Then, v is (1 - x⁴)^(-1/2). To find v', we need to apply the chain rule. The derivative of (something)^(-1/2) is (-1/2)(something)^(-3/2) times the derivative of the something. The something here is (1 - x⁴), so its derivative is -4x³. Therefore, v' = (-1/2)(1 - x⁴)^(-3/2) * (-4x³). Let's compute that step by step.First, multiply (-1/2) by (-4x³). The negatives cancel out, so (-1/2)*(-4x³) = (4x³)/2 = 2x³. Therefore, v' = 2x³*(1 - x⁴)^(-3/2). So, putting that back into the product rule: y' = (1/2)[ (2x)*(1 - x⁴)^(-1/2) + (x² + 2)*(2x³)*(1 - x⁴)^(-3/2) ].Now, let's simplify each term. The first term inside the brackets is 2x times (1 - x⁴)^(-1/2). The second term is (x² + 2) times 2x³ times (1 - x⁴)^(-3/2). So, factoring out common terms might help. Let's see if there's a common factor between the two terms.Notice that both terms have a factor of 2x. The first term is 2x*(1 - x⁴)^(-1/2) and the second term is 2x³*(x² + 2)*(1 - x⁴)^(-3/2). Wait, but the exponents on (1 - x⁴) are different. Maybe we can factor out 2x*(1 - x⁴)^(-3/2) from both terms. Let me check:First term: 2x*(1 - x⁴)^(-1/2) = 2x*(1 - x⁴)^(-1/2) * [ (1 - x⁴)^(-1) / (1 - x⁴)^(-1) ) ] Hmm, maybe that's not the way. Alternatively, to factor out (1 - x⁴)^(-3/2), we need to adjust the first term's exponent. Since (1 - x⁴)^(-1/2) is equal to (1 - x⁴)^(-1/2) * (1 - x⁴)^(-1) divided by (1 - x⁴)^(-1). Wait, maybe this is getting too complicated. Let's try another approach.Let me write both terms with the same exponent. The first term has (1 - x⁴)^(-1/2) and the second term has (1 - x⁴)^(-3/2). So, if I factor out (1 - x⁴)^(-3/2) from both terms, then the first term would have (1 - x⁴)^(-1/2 + 3/2) = (1 - x⁴)^1. Let's see:First term: 2x*(1 - x⁴)^(-1/2) = 2x*(1 - x⁴)^( ( -1/2 + 3/2 ) - 3/2 ) = 2x*(1 - x⁴)^(1 - 3/2) = 2x*(1 - x⁴)^1 * (1 - x⁴)^(-3/2). Wait, maybe that's not helpful. Let's just factor out 2x*(1 - x⁴)^(-3/2) from both terms.First term: 2x*(1 - x⁴)^(-1/2) = 2x*(1 - x⁴)^(-1/2) * 1 = 2x*(1 - x⁴)^(-1/2) * (1 - x⁴)^(-1) / (1 - x⁴)^(-1). Wait, perhaps not. Alternatively, let's factor 2x*(1 - x⁴)^(-3/2) from both terms:First term: 2x*(1 - x⁴)^(-1/2) = 2x*(1 - x⁴)^(-3/2) * (1 - x⁴)^( (-1/2) - (-3/2) ) = 2x*(1 - x⁴)^(-3/2) * (1 - x⁴)^1. Similarly, the second term is 2x³*(x² + 2)*(1 - x⁴)^(-3/2). Therefore, factoring out 2x*(1 - x⁴)^(-3/2) from both terms:First term: 2x*(1 - x⁴)^(-3/2)*(1 - x⁴)Second term: 2x*(1 - x⁴)^(-3/2)*(x² + 2)*x²Wait, let me check:Original first term: 2x*(1 - x⁴)^(-1/2) = 2x*(1 - x⁴)^(-3/2 + 1) = 2x*(1 - x⁴)^(-3/2)*(1 - x⁴)^1Original second term: (x² + 2)*2x³*(1 - x⁴)^(-3/2) = 2x³*(x² + 2)*(1 - x⁴)^(-3/2)So, if we factor out 2x*(1 - x⁴)^(-3/2), then:First term becomes: 2x*(1 - x⁴)^(-3/2)*(1 - x⁴) = 2x*(1 - x⁴)^(-3/2 + 1) = 2x*(1 - x⁴)^(-1/2)Wait, but that's just the original first term again. Maybe this isn't helping. Let me think differently.Alternatively, let's just combine the terms as they are and see if we can factor anything else. So, after computing the derivative:y' = (1/2)[2x*(1 - x⁴)^(-1/2) + 2x³*(x² + 2)*(1 - x⁴)^(-3/2)]We can factor out 2x from both terms inside the brackets:= (1/2)*(2x)[ (1 - x⁴)^(-1/2) + x²*(x² + 2)*(1 - x⁴)^(-3/2) ]The 1/2 and the 2x cancel out to x:= x[ (1 - x⁴)^(-1/2) + x²*(x² + 2)*(1 - x⁴)^(-3/2) ]Now, let's factor out (1 - x⁴)^(-3/2) from both terms inside the brackets. To do that, note that (1 - x⁴)^(-1/2) is equal to (1 - x⁴)^(-3/2 + 1), which is (1 - x⁴)^1 * (1 - x⁴)^(-3/2). Therefore:= x[ (1 - x⁴)*(1 - x⁴)^(-3/2) + x²*(x² + 2)*(1 - x⁴)^(-3/2) ]Factor out (1 - x⁴)^(-3/2):= x*(1 - x⁴)^(-3/2)[ (1 - x⁴) + x²*(x² + 2) ]Now, let's simplify the expression inside the brackets:(1 - x⁴) + x²*(x² + 2) = 1 - x⁴ + x⁴ + 2x² = 1 + 2x²Because the -x⁴ and +x⁴ cancel each other out, leaving 1 + 2x².So now, the derivative becomes:= x*(1 - x⁴)^(-3/2)*(1 + 2x²)We can write this as:= x(1 + 2x²) / (1 - x⁴)^(3/2)Alternatively, if we want to rationalize the denominator or present it differently, but I think this is a simplified form. Let me check if I made any mistakes in the algebra.First, when we factored out (1 - x⁴)^(-3/2), the first term was (1 - x⁴), which is correct. Then the second term was x²*(x² + 2). Adding those together gives 1 - x⁴ + x⁴ + 2x² = 1 + 2x². That seems right.So, the final derivative is x(1 + 2x²) divided by (1 - x⁴)^(3/2). Let me write that in LaTeX to make sure:y' = frac{x(1 + 2x^2)}{(1 - x^4)^{3/2}}Alternatively, since (1 - x⁴) is in the denominator raised to the 3/2 power, it might be written as the square root multiplied by the denominator. But I think the form above is acceptable.Wait, let me check with an alternative approach. Maybe using the quotient rule from the beginning. Let's see:Given y = (x² + 2)/(2√(1 - x⁴)).The quotient rule states that if y = u/v, then y' = (u'v - uv')/v².Let me apply that here. Let u = x² + 2, so u' = 2x. Let v = 2√(1 - x⁴), so we need to find v'. The derivative of 2√(1 - x⁴) is 2*(1/(2√(1 - x⁴)))*(-4x³) = (-4x³)/√(1 - x⁴). Wait, let me compute that again:v = 2*(1 - x⁴)^(1/2). Therefore, v' = 2*(1/2)*(1 - x⁴)^(-1/2)*(-4x³) = (1)*(1 - x⁴)^(-1/2)*(-4x³) = -4x³/(√(1 - x⁴))So, applying the quotient rule:y' = [ (2x)(2√(1 - x⁴)) - (x² + 2)(-4x³)/√(1 - x⁴) ] / [ (2√(1 - x⁴))² ]Simplify numerator and denominator:First, the denominator is [2√(1 - x⁴)]² = 4(1 - x⁴)Numerator:First term: 2x * 2√(1 - x⁴) = 4x√(1 - x⁴)Second term: - (x² + 2) * (-4x³)/√(1 - x⁴) = +4x³(x² + 2)/√(1 - x⁴)Therefore, numerator is 4x√(1 - x⁴) + 4x³(x² + 2)/√(1 - x⁴)To combine these terms, let's get a common denominator. The first term can be written as 4x√(1 - x⁴) * [√(1 - x⁴)/√(1 - x⁴)] = 4x(1 - x⁴)/√(1 - x⁴)Therefore, numerator becomes [4x(1 - x⁴) + 4x³(x² + 2)] / √(1 - x⁴)Factor out 4x from the numerator:4x[ (1 - x⁴) + x²(x² + 2) ] / √(1 - x⁴)Simplify inside the brackets:(1 - x⁴) + x²(x² + 2) = 1 - x⁴ + x⁴ + 2x² = 1 + 2x², same as before.Therefore, numerator is 4x(1 + 2x²)/√(1 - x⁴)Denominator is 4(1 - x⁴)So, y' = [4x(1 + 2x²)/√(1 - x⁴)] / [4(1 - x⁴)] = [4x(1 + 2x²)/√(1 - x⁴)] * [1/(4(1 - x⁴))] = x(1 + 2x²)/[(1 - x⁴) * √(1 - x⁴)] = x(1 + 2x²)/(1 - x⁴)^(3/2)Which matches the result we got earlier. So that's a good check. Therefore, the derivative is indeed y' = x(1 + 2x²)/(1 - x⁴)^(3/2). I should also check if there are any simplifications possible. Let's see, (1 + 2x²) doesn't factor into anything with (1 - x⁴). The denominator is (1 - x⁴)^(3/2), which can be written as (1 - x²)^(3/2)(1 + x²)^(3/2), but I don't think that helps. So, the answer is as simplified as it can be.Another way to present it might be:y' = frac{x(1 + 2x^2)}{(1 - x^4)sqrt{1 - x^4}}But since (1 - x⁴)^(3/2) is the same as (1 - x⁴) multiplied by sqrt(1 - x⁴), both forms are equivalent.Therefore, the final answer is x(1 + 2x²) divided by (1 - x⁴) raised to the 3/2 power. So, in boxed form, that would be:boxed{dfrac{x(1 + 2x^{2})}{(1 - x^{4})^{3/2}}}</think>To find the derivative of the function ( y = frac{x^{2} + 2}{2 sqrt{1 - x^{4}}} ), we can use the product rule and the chain rule.First, rewrite the function as:[ y = frac{1}{2} (x^2 + 2)(1 - x^4)^{-1/2} ]Using the product rule, where ( u = x^2 + 2 ) and ( v = (1 - x^4)^{-1/2} ):[ y' = frac{1}{2} left[ u'v + uv' right] ]Compute ( u' ) and ( v' ):- ( u' = 2x )- ( v' = frac{d}{dx} (1 - x^4)^{-1/2} = -frac{1}{2} (1 - x^4)^{-3/2} (-4x^3) = 2x^3 (1 - x^4)^{-3/2} )Substitute ( u' ), ( v ), and ( v' ) into the product rule:[ y' = frac{1}{2} left[ 2x (1 - x^4)^{-1/2} + (x^2 + 2) 2x^3 (1 - x^4)^{-3/2} right] ]Factor out common terms:[ y' = frac{1}{2} cdot 2x left[ (1 - x^4)^{-1/2} + x^2 (x^2 + 2) (1 - x^4)^{-3/2} right] ][ y' = x left[ (1 - x^4)^{-1/2} + x^2 (x^2 + 2) (1 - x^4)^{-3/2} right] ]Factor out ( (1 - x^4)^{-3/2} ):[ y' = x (1 - x^4)^{-3/2} left[ (1 - x^4) + x^2 (x^2 + 2) right] ]Simplify the expression inside the brackets:[ (1 - x^4) + x^2 (x^2 + 2) = 1 - x^4 + x^4 + 2x^2 = 1 + 2x^2 ]Thus, the derivative is:[ y' = frac{x (1 + 2x^2)}{(1 - x^4)^{3/2}} ]The final answer is:[ boxed{dfrac{x(1 + 2x^{2})}{(1 - x^{4})^{3/2}}} ]]

🔑:## Solutiony^{prime}=left(frac{x^{2}+2}{2 sqrt{1-x^{4}}}right)^{prime}=frac{2 x cdot sqrt{1-x^{4}}-left(x^{2}+2right) cdot frac{1}{2 sqrt{1-x^{4}}} cdotleft(-4 x^{3}right)}{2left(1-x^{4}right)}==frac{x cdotleft(1-x^{4}right)+x^{3}left(x^{2}+2right)}{left(1-x^{4}right) sqrt{1-x^{4}}}=frac{x-x^{5}+x^{5}+2 x^{3}}{left(1-x^{4}right) sqrt{1-x^{4}}}=frac{2 x^{3}+x}{left(1-x^{4}right) sqrt{1-x^{4}}}Kuznetsov Differentiation 6-28