🧠:<think>Okay, so I have these two problems to solve related to a music streaming app. Let me take them one by one.Starting with the first problem. It's about Fourier series and energy concentration. The app models the sound quality of vinyl records as a continuous function f(t), where t is time in seconds, and f(t) is the amplitude. The goal is to replicate the rich harmonic content by ensuring that at least 95% of the energy is in the first 10 harmonics.Hmm, I remember that the energy of a signal can be related to its Fourier series coefficients through Parseval's theorem. Parseval's theorem states that the total energy of a function is equal to the sum of the squares of its Fourier coefficients. So, if the function f(t) has a Fourier series expansion, the energy E is the integral of |f(t)|² dt from 0 to T, which equals the sum of the squares of the Fourier coefficients.Wait, more precisely, for a periodic function f(t) with period T, the Fourier series is given by:f(t) = a₀/2 + Σ (a_n cos(nω₀ t) + b_n sin(nω₀ t))where ω₀ = 2π/T. The Fourier coefficients are a₀, a_n, and b_n.According to Parseval's theorem, the total energy E is:E = (a₀²)/4 + (1/2) Σ (a_n² + b_n²) for n=1 to ∞.So, the total energy is the sum of the squares of the Fourier coefficients, each multiplied by 1/2 except for a₀ which is multiplied by 1/4.The problem wants at least 95% of the energy concentrated in the first 10 harmonics. So, the energy in the first 10 harmonics should be 0.95E.Let me denote the energy in the first 10 harmonics as E_10. Then,E_10 = (a₀²)/4 + (1/2) Σ (a_n² + b_n²) for n=1 to 10.And we need E_10 / E ≥ 0.95.So, substituting E from Parseval's theorem:[(a₀²)/4 + (1/2) Σ (a_n² + b_n²) from n=1 to 10] / [(a₀²)/4 + (1/2) Σ (a_n² + b_n²) from n=1 to ∞] ≥ 0.95.That's the condition that must be satisfied by the Fourier coefficients a_n and b_n.Wait, but the problem says "derive the condition that must be satisfied by the Fourier coefficients a_n and b_n". So, essentially, the ratio of the energy in the first 10 harmonics to the total energy must be at least 95%.So, the condition is:[(a₀²)/4 + (1/2)(a₁² + b₁² + a₂² + b₂² + ... + a₁₀² + b₁₀²)] / [(a₀²)/4 + (1/2)(a₁² + b₁² + a₂² + b₂² + ...)] ≥ 0.95.Alternatively, we can write it as:Σ_{n=0}^{10} (c_n²) / Σ_{n=0}^{∞} (c_n²) ≥ 0.95,where c_n are the Fourier coefficients, considering that c₀ = a₀/2 and c_n = (a_n + ib_n)/2 for n ≥ 1.But in the problem, they specifically mention a_n and b_n, so probably better to stick with the expression in terms of a_n and b_n.So, the condition is that the sum of the squares of the first 10 Fourier coefficients (including a₀) divided by the total sum of squares of all coefficients must be at least 0.95.Moving on to the second problem. It's about using Singular Value Decomposition (SVD) to form a low-rank approximation of a matrix A that represents user listening habits. The matrix A is m x n, with rank r. The goal is to find a low-rank approximation A_k that retains 90% of the total variance.I remember that SVD decomposes a matrix A into A = UΣV^T, where U and V are orthogonal matrices, and Σ is a diagonal matrix of singular values. The singular values are in descending order, so the first singular value is the largest.The total variance in the data is the sum of the squares of the singular values. So, to retain 90% of the variance, we need to find the smallest k such that the sum of the squares of the first k singular values divided by the total sum is at least 0.90.But the problem says to express how the engineer can use the SVD of A to form a low-rank approximation A_k that retains 90% of the total variance. It also asks to specify the number k of singular values needed in terms of r, and explain the steps to compute A_k.Wait, but the rank of A is r. So, the number of non-zero singular values is r. So, if we take k singular values, k must be less than or equal to r.But the problem says to retain 90% of the total variance. So, the engineer would compute the SVD of A, get the singular values, compute the cumulative sum of their squares, and find the smallest k such that the cumulative sum divided by the total sum is at least 0.90.But the problem asks to express this in terms of r. Hmm, but r is the rank, which is the number of non-zero singular values. So, if the matrix has rank r, then all singular values beyond r are zero. So, to get 90% variance, k would be some number less than or equal to r.But unless we know more about the distribution of the singular values, we can't specify k in terms of r without additional information. For example, if the singular values decay rapidly, k might be much smaller than r. But if they decay slowly, k might be close to r.Wait, but the problem says "express how the engineer can use the SVD of A to form a low-rank approximation A_k that retains 90% of the total variance. Specify the number k of singular values needed in terms of r, and explain the steps needed to compute A_k."Hmm, maybe it's expecting that k is the smallest integer such that the sum of the first k singular values squared divided by the total sum is ≥ 0.90. So, in terms of r, k would be ≤ r.But perhaps the answer is that k is the number of singular values needed such that the cumulative variance is 90%, which would be found by summing the squares of the singular values until 90% is reached, and k is the number needed. But since the rank is r, k can't exceed r.Alternatively, maybe the problem expects that k is the number such that the sum of the first k singular values squared is 90% of the total sum, and since the total sum is the sum of all r singular values squared, k is the smallest integer where the cumulative sum up to k is ≥ 0.9 times the total.But the problem says "specify the number k of singular values needed in terms of r". Hmm, unless it's implying that k is a certain fraction of r, but I don't think so. It's more likely that k is determined by the cumulative variance, regardless of r, but constrained by r.Wait, but the rank is r, so the maximum k is r. So, the steps would be:1. Compute the SVD of A, obtaining U, Σ, V^T.2. Compute the total variance as the sum of the squares of the singular values, which is the trace of Σ².3. Compute the cumulative sum of the squares of the singular values in descending order.4. Find the smallest k such that the cumulative sum up to k is at least 90% of the total variance.5. Then, A_k is formed by taking the first k columns of U, the first k singular values, and the first k rows of V^T, so A_k = U_k Σ_k V_k^T.But the problem asks to specify k in terms of r. Since r is the rank, and the singular values beyond r are zero, k must be ≤ r. But without knowing the distribution, we can't express k as a function of r. So perhaps the answer is that k is the smallest integer such that the sum of the first k singular values squared is at least 0.9 times the total sum, and k ≤ r.Alternatively, maybe the problem expects that k is the number of singular values needed, which is determined by the cumulative variance, and since the rank is r, k can be up to r, but in practice, it's less.Wait, maybe the problem is expecting that since the total variance is the sum of all r singular values squared, then k is the number such that the sum of the first k singular values squared divided by the total sum is ≥ 0.9. So, k is the minimal integer where this holds, and k ≤ r.So, in terms of r, k is the minimal integer such that Σ_{i=1}^k σ_i² / Σ_{i=1}^r σ_i² ≥ 0.9.Therefore, the steps are:1. Perform SVD on A to get U, Σ, V^T.2. Compute the total variance as the sum of squares of all singular values (which is the sum of the diagonal elements of Σ squared).3. Compute the cumulative sum of the squares of the singular values in descending order.4. Find the smallest k such that the cumulative sum up to k is at least 90% of the total variance.5. The low-rank approximation A_k is then U_k Σ_k V_k^T, where U_k consists of the first k columns of U, Σ_k is a k x k diagonal matrix with the first k singular values, and V_k^T consists of the first k rows of V^T.So, in terms of r, k is the minimal integer where the cumulative variance reaches 90%, and k ≤ r.I think that's the approach.</think>

🧠:<think>Alright, so I've got this problem about a state senator proposing a tax cut to attract businesses and boost the economy. It's split into two parts, and I need to figure out the new total corporate tax revenue and then determine the range for the tax cut to ensure revenue doesn't drop below a certain percentage. Let me try to break this down step by step.Starting with part 1: I need to derive an expression for the new total corporate tax revenue ( R_1 ) after the tax cut. The current tax rate is ( r_0 ), and the proposed rate is ( r_1 = r_0 - Delta r ). The number of businesses is expected to increase from ( N_0 ) to ( N_1 = N_0 cdot (1 + alpha cdot Delta r) ). The current revenue ( R_0 ) is given as ( N_0 cdot frac{r_0}{100} cdot B ), where ( B ) is the average taxable income per business.Okay, so for ( R_1 ), it should be similar to ( R_0 ), but with the new number of businesses and the new tax rate. So, ( R_1 = N_1 cdot frac{r_1}{100} cdot B ). Let me write that out:( R_1 = N_1 cdot frac{r_1}{100} cdot B )Substituting the expressions for ( N_1 ) and ( r_1 ):( R_1 = N_0 cdot (1 + alpha cdot Delta r) cdot frac{r_0 - Delta r}{100} cdot B )Hmm, that seems right. Let me make sure I didn't miss anything. The number of businesses increases by a factor dependent on ( Delta r ), and the tax rate decreases by ( Delta r ). So, multiplying those together with ( N_0 ) and the other constants should give the new revenue.So, simplifying this, I can write:( R_1 = N_0 cdot B cdot frac{(1 + alpha Delta r)(r_0 - Delta r)}{100} )I think that's the expression for ( R_1 ). Let me just double-check the substitution. Yes, ( N_1 ) is substituted correctly, and ( r_1 ) is substituted as ( r_0 - Delta r ). So that should be correct.Moving on to part 2: The senator wants ( R_1 ) to be at least ( p % ) of ( R_0 ). So, we need ( R_1 geq frac{p}{100} R_0 ). I need to find the range of ( Delta r ) that satisfies this inequality.First, let's write the inequality:( R_1 geq frac{p}{100} R_0 )Substituting the expressions for ( R_1 ) and ( R_0 ):( N_0 cdot B cdot frac{(1 + alpha Delta r)(r_0 - Delta r)}{100} geq frac{p}{100} cdot N_0 cdot frac{r_0}{100} cdot B )Wait, hold on. Let me make sure I substitute ( R_0 ) correctly. ( R_0 = N_0 cdot frac{r_0}{100} cdot B ), so ( frac{p}{100} R_0 = frac{p}{100} cdot N_0 cdot frac{r_0}{100} cdot B ). So, the inequality becomes:( N_0 cdot B cdot frac{(1 + alpha Delta r)(r_0 - Delta r)}{100} geq frac{p}{100} cdot N_0 cdot frac{r_0}{100} cdot B )I notice that ( N_0 ) and ( B ) are on both sides, so we can divide both sides by ( N_0 cdot B ) to simplify:( frac{(1 + alpha Delta r)(r_0 - Delta r)}{100} geq frac{p}{100} cdot frac{r_0}{100} )Simplify further by multiplying both sides by 100:( (1 + alpha Delta r)(r_0 - Delta r) geq frac{p}{100} r_0 )So, now we have:( (1 + alpha Delta r)(r_0 - Delta r) geq frac{p}{100} r_0 )Let me expand the left side:( (1)(r_0) + (1)(- Delta r) + (alpha Delta r)(r_0) + (alpha Delta r)(- Delta r) geq frac{p}{100} r_0 )Simplify term by term:1. ( 1 cdot r_0 = r_0 )2. ( 1 cdot (- Delta r) = - Delta r )3. ( alpha Delta r cdot r_0 = alpha r_0 Delta r )4. ( alpha Delta r cdot (- Delta r) = - alpha (Delta r)^2 )Putting it all together:( r_0 - Delta r + alpha r_0 Delta r - alpha (Delta r)^2 geq frac{p}{100} r_0 )Now, let's bring all terms to the left side to set up the inequality for solving:( r_0 - Delta r + alpha r_0 Delta r - alpha (Delta r)^2 - frac{p}{100} r_0 geq 0 )Factor out ( r_0 ) where possible:( r_0 (1 - frac{p}{100}) - Delta r + alpha r_0 Delta r - alpha (Delta r)^2 geq 0 )Let me rearrange the terms for clarity:( - alpha (Delta r)^2 + (alpha r_0 - 1) Delta r + r_0 (1 - frac{p}{100}) geq 0 )This is a quadratic inequality in terms of ( Delta r ). Let me write it as:( - alpha (Delta r)^2 + (alpha r_0 - 1) Delta r + r_0 (1 - frac{p}{100}) geq 0 )To make it a bit more standard, I can multiply both sides by -1, which will reverse the inequality sign:( alpha (Delta r)^2 + (- alpha r_0 + 1) Delta r - r_0 (1 - frac{p}{100}) leq 0 )So, now we have:( alpha (Delta r)^2 + (1 - alpha r_0) Delta r - r_0 (1 - frac{p}{100}) leq 0 )Let me denote this quadratic as ( Q(Delta r) leq 0 ), where:( Q(Delta r) = alpha (Delta r)^2 + (1 - alpha r_0) Delta r - r_0 (1 - frac{p}{100}) )To find the values of ( Delta r ) that satisfy this inequality, we need to find the roots of the quadratic equation ( Q(Delta r) = 0 ) and determine the intervals where the quadratic is less than or equal to zero.First, let's write the quadratic equation:( alpha (Delta r)^2 + (1 - alpha r_0) Delta r - r_0 (1 - frac{p}{100}) = 0 )This is a quadratic in the form ( a x^2 + b x + c = 0 ), where:- ( a = alpha )- ( b = 1 - alpha r_0 )- ( c = - r_0 (1 - frac{p}{100}) )The solutions to this quadratic can be found using the quadratic formula:( Delta r = frac{ -b pm sqrt{b^2 - 4ac} }{2a} )Plugging in the values:( Delta r = frac{ - (1 - alpha r_0) pm sqrt{(1 - alpha r_0)^2 - 4 cdot alpha cdot (- r_0 (1 - frac{p}{100}))} }{2 alpha} )Simplify the discriminant ( D ):( D = (1 - alpha r_0)^2 - 4 cdot alpha cdot (- r_0 (1 - frac{p}{100})) )Calculate each part:First, expand ( (1 - alpha r_0)^2 ):( 1 - 2 alpha r_0 + alpha^2 r_0^2 )Then, compute the second term:( -4 alpha (- r_0 (1 - frac{p}{100})) = 4 alpha r_0 (1 - frac{p}{100}) )So, the discriminant becomes:( D = 1 - 2 alpha r_0 + alpha^2 r_0^2 + 4 alpha r_0 (1 - frac{p}{100}) )Let me simplify this:First, distribute the 4αr0:( 4 alpha r_0 (1 - frac{p}{100}) = 4 alpha r_0 - frac{4 alpha r_0 p}{100} )So, putting it all together:( D = 1 - 2 alpha r_0 + alpha^2 r_0^2 + 4 alpha r_0 - frac{4 alpha r_0 p}{100} )Combine like terms:- The constant term is 1.- The terms with ( alpha r_0 ): ( -2 alpha r_0 + 4 alpha r_0 = 2 alpha r_0 )- The term with ( alpha^2 r_0^2 ): ( alpha^2 r_0^2 )- The term with ( alpha r_0 p ): ( - frac{4 alpha r_0 p}{100} )So, putting it together:( D = 1 + 2 alpha r_0 + alpha^2 r_0^2 - frac{4 alpha r_0 p}{100} )Hmm, let me factor this expression if possible. It looks like a quadratic in ( alpha r_0 ), but I'm not sure. Alternatively, maybe I can write it as:( D = (alpha r_0)^2 + 2 alpha r_0 (1 - frac{2 p}{100}) + 1 )Wait, let me check:( (alpha r_0)^2 + 2 alpha r_0 (1 - frac{2 p}{100}) + 1 )Expanding this:( alpha^2 r_0^2 + 2 alpha r_0 - frac{4 alpha r_0 p}{100} + 1 )Which is exactly what we have for D. So, yes, we can write:( D = (alpha r_0)^2 + 2 alpha r_0 (1 - frac{2 p}{100}) + 1 )Alternatively, maybe it's a perfect square? Let me see:Let me denote ( x = alpha r_0 ). Then,( D = x^2 + 2 x (1 - frac{2 p}{100}) + 1 )This is a quadratic in x. Let me see if it can be expressed as a square:( x^2 + 2 x (1 - frac{2 p}{100}) + (1 - frac{2 p}{100})^2 - (1 - frac{2 p}{100})^2 + 1 )Wait, that might complicate things. Alternatively, maybe I can just proceed with the discriminant as it is.So, going back to the quadratic formula:( Delta r = frac{ - (1 - alpha r_0) pm sqrt{D} }{2 alpha} )Where ( D = 1 + 2 alpha r_0 + alpha^2 r_0^2 - frac{4 alpha r_0 p}{100} )Hmm, this is getting a bit messy. Maybe I can factor D differently or see if it's a perfect square.Wait, let me think again. The discriminant is:( D = (alpha r_0)^2 + 2 alpha r_0 (1 - frac{2 p}{100}) + 1 )Which is:( D = (alpha r_0 + (1 - frac{2 p}{100}))^2 - (1 - frac{2 p}{100})^2 + 1 )Wait, let me compute:( (alpha r_0 + a)^2 = alpha^2 r_0^2 + 2 a alpha r_0 + a^2 )Comparing to D:( alpha^2 r_0^2 + 2 alpha r_0 (1 - frac{2 p}{100}) + 1 )So, if I set ( a = 1 - frac{2 p}{100} ), then:( (alpha r_0 + a)^2 = alpha^2 r_0^2 + 2 a alpha r_0 + a^2 )But D is:( alpha^2 r_0^2 + 2 a alpha r_0 + 1 )So, the difference is ( a^2 ) vs 1. So,( D = (alpha r_0 + a)^2 - (a^2 - 1) )Which is:( D = (alpha r_0 + a)^2 - (a^2 - 1) )But I don't know if this helps. Maybe not. Let me just proceed with the quadratic formula.So, the roots are:( Delta r = frac{ - (1 - alpha r_0) pm sqrt{D} }{2 alpha} )Let me compute the numerator:( - (1 - alpha r_0) = -1 + alpha r_0 )So,( Delta r = frac{ -1 + alpha r_0 pm sqrt{D} }{2 alpha} )So, the two roots are:( Delta r_1 = frac{ -1 + alpha r_0 + sqrt{D} }{2 alpha} )and( Delta r_2 = frac{ -1 + alpha r_0 - sqrt{D} }{2 alpha} )Now, since the quadratic coefficient ( a = alpha ) is positive (assuming ( alpha > 0 ), which makes sense as a sensitivity coefficient), the parabola opens upwards. Therefore, the quadratic ( Q(Delta r) ) is less than or equal to zero between its two roots.So, the solution to ( Q(Delta r) leq 0 ) is ( Delta r ) between ( Delta r_2 ) and ( Delta r_1 ).But we need to ensure that ( Delta r ) is positive because it's a tax cut, so ( Delta r > 0 ). Also, the new tax rate ( r_1 = r_0 - Delta r ) must be positive, so ( Delta r < r_0 ). Therefore, ( 0 < Delta r < r_0 ).So, the allowable range for ( Delta r ) is between ( Delta r_2 ) and ( Delta r_1 ), but we need to check if these roots lie within the interval ( (0, r_0) ).But since the problem states that ( 0 < p < 100 ), and we're dealing with a tax cut that should not reduce revenue below ( p % ) of the original, it's likely that there will be a valid range for ( Delta r ).However, calculating the exact roots might be complicated, so perhaps we can express the range in terms of the roots.Therefore, the range of ( Delta r ) is:( Delta r_2 leq Delta r leq Delta r_1 )But since ( Delta r ) must be positive and less than ( r_0 ), we have to consider the intersection of these intervals.But perhaps, instead of trying to write the exact roots, we can express the range in terms of the quadratic inequality.Alternatively, maybe I can rearrange the quadratic inequality:( alpha (Delta r)^2 + (1 - alpha r_0) Delta r - r_0 (1 - frac{p}{100}) leq 0 )Let me denote ( x = Delta r ) for simplicity:( alpha x^2 + (1 - alpha r_0) x - r_0 (1 - frac{p}{100}) leq 0 )We can solve for x:The solutions are between the two roots ( x_1 ) and ( x_2 ), where ( x_1 < x_2 ). Since the coefficient of ( x^2 ) is positive, the quadratic is positive outside the interval ( [x_1, x_2] ) and negative inside.Therefore, the inequality ( alpha x^2 + (1 - alpha r_0) x - r_0 (1 - frac{p}{100}) leq 0 ) holds for ( x ) between ( x_1 ) and ( x_2 ).But since ( x = Delta r ) must be positive and less than ( r_0 ), the allowable range is the intersection of ( [x_1, x_2] ) with ( (0, r_0) ).Therefore, the range of ( Delta r ) is:( max(0, x_1) leq Delta r leq min(r_0, x_2) )But without knowing the exact values of ( alpha ), ( r_0 ), and ( p ), we can't simplify further. However, we can express the range in terms of the roots.Alternatively, perhaps we can write the inequality in a different form.Wait, going back to the inequality before expanding:( (1 + alpha Delta r)(r_0 - Delta r) geq frac{p}{100} r_0 )Maybe we can solve this inequality without expanding, but I think expanding was the right approach.Alternatively, let me consider that ( Delta r ) is small compared to ( r_0 ), but I don't think we can assume that.Alternatively, maybe we can write the inequality as:( (1 + alpha Delta r)(r_0 - Delta r) geq k r_0 ), where ( k = frac{p}{100} )So,( (1 + alpha Delta r)(r_0 - Delta r) geq k r_0 )Divide both sides by ( r_0 ):( (1 + alpha Delta r)left(1 - frac{Delta r}{r_0}right) geq k )Let me denote ( x = frac{Delta r}{r_0} ), so ( 0 < x < 1 ) because ( Delta r < r_0 ).Then, the inequality becomes:( (1 + alpha r_0 x)left(1 - xright) geq k )Expanding this:( (1)(1 - x) + alpha r_0 x (1 - x) geq k )Which is:( 1 - x + alpha r_0 x - alpha r_0 x^2 geq k )Rearranging:( - alpha r_0 x^2 + (alpha r_0 - 1) x + 1 - k geq 0 )Multiply both sides by -1 (reversing the inequality):( alpha r_0 x^2 + (- alpha r_0 + 1) x + (k - 1) leq 0 )So, now we have a quadratic in x:( alpha r_0 x^2 + (1 - alpha r_0) x + (k - 1) leq 0 )This is similar to what we had before, but now in terms of x, which is ( Delta r / r_0 ).Let me denote this quadratic as ( Q(x) leq 0 ), where:( Q(x) = alpha r_0 x^2 + (1 - alpha r_0) x + (k - 1) )Again, solving ( Q(x) = 0 ) will give the critical points.Using the quadratic formula:( x = frac{ - (1 - alpha r_0) pm sqrt{(1 - alpha r_0)^2 - 4 cdot alpha r_0 cdot (k - 1)} }{2 alpha r_0} )Simplify the discriminant:( D = (1 - alpha r_0)^2 - 4 alpha r_0 (k - 1) )Expanding:( D = 1 - 2 alpha r_0 + alpha^2 r_0^2 - 4 alpha r_0 k + 4 alpha r_0 )Simplify:Combine like terms:- Constant term: 1- Terms with ( alpha r_0 ): ( -2 alpha r_0 + 4 alpha r_0 = 2 alpha r_0 )- Terms with ( alpha^2 r_0^2 ): ( alpha^2 r_0^2 )- Terms with ( alpha r_0 k ): ( -4 alpha r_0 k )So,( D = 1 + 2 alpha r_0 + alpha^2 r_0^2 - 4 alpha r_0 k )Wait, this looks similar to what we had earlier. Indeed, since ( k = frac{p}{100} ), this is consistent.So, the roots are:( x = frac{ - (1 - alpha r_0) pm sqrt{1 + 2 alpha r_0 + alpha^2 r_0^2 - 4 alpha r_0 k} }{2 alpha r_0} )Simplify numerator:( -1 + alpha r_0 pm sqrt{1 + 2 alpha r_0 + alpha^2 r_0^2 - 4 alpha r_0 k} )So, the roots are:( x_1 = frac{ -1 + alpha r_0 - sqrt{1 + 2 alpha r_0 + alpha^2 r_0^2 - 4 alpha r_0 k} }{2 alpha r_0} )and( x_2 = frac{ -1 + alpha r_0 + sqrt{1 + 2 alpha r_0 + alpha^2 r_0^2 - 4 alpha r_0 k} }{2 alpha r_0} )Since ( x = Delta r / r_0 ), the allowable range for ( x ) is between ( x_1 ) and ( x_2 ), but considering ( x > 0 ) and ( x < 1 ).Therefore, the allowable range for ( Delta r ) is:( r_0 x_1 leq Delta r leq r_0 x_2 )But since ( x_1 ) might be negative, we take the maximum of ( r_0 x_1 ) and 0, and the minimum of ( r_0 x_2 ) and ( r_0 ).However, without specific values, it's hard to simplify further. So, perhaps the best way to present the range is in terms of the roots.Alternatively, maybe we can express the range as:( Delta r leq frac{ - (1 - alpha r_0) + sqrt{(1 - alpha r_0)^2 + 4 alpha r_0 (1 - frac{p}{100})} }{2 alpha} )But this seems similar to what we had before.Wait, let me think differently. Maybe instead of solving the quadratic, we can rearrange the inequality.Starting from:( (1 + alpha Delta r)(r_0 - Delta r) geq frac{p}{100} r_0 )Let me divide both sides by ( r_0 ):( (1 + alpha Delta r)left(1 - frac{Delta r}{r_0}right) geq frac{p}{100} )Let me denote ( x = Delta r ), so:( (1 + alpha x)left(1 - frac{x}{r_0}right) geq frac{p}{100} )Expanding:( 1 + alpha x - frac{x}{r_0} - frac{alpha x^2}{r_0} geq frac{p}{100} )Rearranging:( - frac{alpha x^2}{r_0} + (alpha - frac{1}{r_0}) x + 1 - frac{p}{100} geq 0 )Multiply both sides by ( - r_0 / alpha ) (which is negative, so inequality sign flips):( x^2 - left( alpha r_0 - 1 right) x - frac{r_0}{alpha} left( 1 - frac{p}{100} right) leq 0 )So, now we have:( x^2 - (alpha r_0 - 1) x - frac{r_0}{alpha} left( 1 - frac{p}{100} right) leq 0 )This is another quadratic in x, which is ( Delta r ). Let me denote this as:( Q(x) = x^2 - (alpha r_0 - 1) x - frac{r_0}{alpha} left( 1 - frac{p}{100} right) leq 0 )The roots of ( Q(x) = 0 ) are:( x = frac{ (alpha r_0 - 1) pm sqrt{ (alpha r_0 - 1)^2 + 4 cdot frac{r_0}{alpha} left( 1 - frac{p}{100} right) } }{2} )Simplify the discriminant:( D = (alpha r_0 - 1)^2 + 4 cdot frac{r_0}{alpha} left( 1 - frac{p}{100} right) )Expanding ( (alpha r_0 - 1)^2 ):( alpha^2 r_0^2 - 2 alpha r_0 + 1 )So,( D = alpha^2 r_0^2 - 2 alpha r_0 + 1 + frac{4 r_0}{alpha} left( 1 - frac{p}{100} right) )This is the same discriminant as before, just expressed differently.So, the roots are:( x = frac{ (alpha r_0 - 1) pm sqrt{ alpha^2 r_0^2 - 2 alpha r_0 + 1 + frac{4 r_0}{alpha} left( 1 - frac{p}{100} right) } }{2} )Again, this is quite complex. I think it's best to leave the range in terms of the roots, as solving for ( Delta r ) explicitly would require knowing the specific values of ( alpha ), ( r_0 ), and ( p ).Therefore, the range of ( Delta r ) is between the two roots of the quadratic equation derived from the inequality ( R_1 geq frac{p}{100} R_0 ). Specifically, ( Delta r ) must satisfy:( Delta r_2 leq Delta r leq Delta r_1 )Where ( Delta r_1 ) and ( Delta r_2 ) are the roots given by:( Delta r = frac{ - (1 - alpha r_0) pm sqrt{(1 - alpha r_0)^2 + 4 alpha r_0 (1 - frac{p}{100})} }{2 alpha} )But considering the physical constraints ( 0 < Delta r < r_0 ), the allowable range is the intersection of this interval with the roots.Alternatively, since the quadratic opens upwards (as ( alpha > 0 )), the inequality ( Q(Delta r) leq 0 ) holds between the two roots. Therefore, the allowable ( Delta r ) must be between the smaller root ( Delta r_2 ) and the larger root ( Delta r_1 ).However, since ( Delta r ) must be positive and less than ( r_0 ), the actual range is:( maxleft(0, Delta r_2right) leq Delta r leq minleft(r_0, Delta r_1right) )But without specific values, we can't simplify this further. Therefore, the range of ( Delta r ) is between the two roots of the quadratic equation, considering the constraints ( 0 < Delta r < r_0 ).So, summarizing:1. The new revenue ( R_1 ) is ( N_0 cdot B cdot frac{(1 + alpha Delta r)(r_0 - Delta r)}{100} ).2. The allowable ( Delta r ) must satisfy the quadratic inequality, resulting in a range between the two roots, adjusted for positivity and ( Delta r < r_0 ).I think that's as far as I can go without specific numerical values. So, the final answer for part 1 is the expression for ( R_1 ), and for part 2, it's the range of ( Delta r ) between the two roots, considering the constraints.</think>

🧠:<think>Okay, so I have this problem about PCA and its sensitivity and interpretability. Let me try to work through it step by step. First, the problem is divided into two parts. Part 1 is about eigenvalue sensitivity analysis, and Part 2 is about interpretability and reconstruction error when the data might lie on a non-linear manifold. Let me tackle them one by one.Part 1: Eigenvalue Sensitivity AnalysisAlright, so we have a covariance matrix Σ with eigenvalues λ₁ ≥ λ₂ ≥ ... ≥ λ_p. We're looking at the proportion of variance explained by the first k principal components, which is P_k = (Σ_{i=1}^k λ_i) / (Σ_{j=1}^p λ_j). The task is to analyze how sensitive P_k is to small perturbations in the eigenvalues. Specifically, if each λ_i is perturbed by a small ε_i, with |ε_i| << λ_i, we need to derive the change in P_k.Hmm, okay. So, let's denote the perturbed eigenvalues as λ_i' = λ_i + ε_i. Then, the perturbed proportion P_k' would be (Σ_{i=1}^k (λ_i + ε_i)) / (Σ_{j=1}^p (λ_j + ε_j)).We need to find the change ΔP_k = P_k' - P_k.Let me compute this:ΔP_k = [ (Σ_{i=1}^k λ_i + Σ_{i=1}^k ε_i) / (Σ_{j=1}^p λ_j + Σ_{j=1}^p ε_j) ] - [ Σ_{i=1}^k λ_i / Σ_{j=1}^p λ_j ]Let me denote S = Σ_{j=1}^p λ_j, which is the total variance. Similarly, let T = Σ_{i=1}^k λ_i, so P_k = T/S.Similarly, the perturbed total variance is S' = S + Σ_{j=1}^p ε_j, and the perturbed T' = T + Σ_{i=1}^k ε_i.So, P_k' = T' / S' = (T + ΔT) / (S + ΔS), where ΔT = Σ_{i=1}^k ε_i and ΔS = Σ_{j=1}^p ε_j.So, ΔP_k = (T + ΔT)/(S + ΔS) - T/S.To find this difference, let's write it as:ΔP_k = [ (T + ΔT)S - T(S + ΔS) ] / [S(S + ΔS)]Simplify numerator:(T + ΔT)S - T(S + ΔS) = TS + ΔT S - TS - T ΔS = ΔT S - T ΔSSo, ΔP_k = (ΔT S - T ΔS) / [S(S + ΔS)]Since |ε_i| << λ_i, the perturbations are small, so ΔS and ΔT are small compared to S and T respectively. Therefore, S + ΔS ≈ S, so the denominator is approximately S^2.Thus, ΔP_k ≈ (ΔT S - T ΔS) / S^2 = (ΔT / S) - (T ΔS) / S^2But T/S is P_k, so this becomes:ΔP_k ≈ (ΔT / S) - (P_k ΔS / S)Which can be written as:ΔP_k ≈ (1/S) ΔT - (P_k / S) ΔSBut ΔT = Σ_{i=1}^k ε_i and ΔS = Σ_{j=1}^p ε_j.So, substituting back:ΔP_k ≈ (1/S)(Σ_{i=1}^k ε_i) - (P_k / S)(Σ_{j=1}^p ε_j)Alternatively, factoring out 1/S:ΔP_k ≈ (1/S)[ Σ_{i=1}^k ε_i - P_k Σ_{j=1}^p ε_j ]Hmm, that seems right. Let me check the dimensions. Each term is a sum of ε's scaled by 1/S, which makes sense because P_k is a proportion.Alternatively, we can write it as:ΔP_k ≈ (1/S) Σ_{i=1}^k ε_i - (P_k / S) Σ_{j=1}^p ε_jWhich is the same as:ΔP_k ≈ (1/S) [ Σ_{i=1}^k ε_i - P_k Σ_{j=1}^p ε_j ]Yes, that looks correct. So, the change in P_k is approximately the average of the perturbations in the first k eigenvalues minus P_k times the average perturbation across all eigenvalues.Wait, actually, it's scaled by 1/S, which is the inverse of the total variance. So, it's the sum of the perturbations in the first k eigenvalues divided by total variance, minus P_k times the sum of all perturbations divided by total variance.Alternatively, we can factor out 1/S:ΔP_k ≈ (1/S) [ Σ_{i=1}^k ε_i - P_k Σ_{j=1}^p ε_j ]Which is the expression we have.So, that's the sensitivity of P_k to small perturbations in the eigenvalues.Part 2: Interpretability and Reconstruction ErrorNow, this part is about the limitations of PCA when the data lies on a non-linear manifold. The question is, if the true structure is non-linear, how does PCA's reconstruction error compare when we have an ideal non-linear method with zero reconstruction error.Given that the intrinsic dimensionality is d < k < p, and we're using a linear k-dimensional PCA. We need to find the conditions under which PCA's reconstruction error is minimized.First, let's recall that PCA minimizes the reconstruction error in the least squares sense. That is, among all k-dimensional linear subspaces, PCA finds the one that minimizes the sum of squared reconstruction errors.But if the data lies on a non-linear manifold of dimension d < k, then the PCA might not capture the true structure as well as a non-linear method. However, since k > d, PCA might still capture some of the variance, but perhaps not optimally.Wait, but in this case, the non-linear method has zero reconstruction error, meaning it perfectly captures the data structure. So, the PCA, being linear, might have some reconstruction error even though k > d.But the question is, under what conditions is PCA's reconstruction error minimized?Hmm. Let me think.Reconstruction error in PCA is the sum of squared distances from each data point to its projection onto the PCA subspace. If the data lies on a non-linear manifold, the PCA subspace might not align well with the manifold, leading to higher reconstruction error.But since k > d, the PCA subspace has higher dimensionality than the intrinsic manifold. So, theoretically, a higher-dimensional linear space can approximate a lower-dimensional non-linear manifold better, but it's not guaranteed.Wait, but if the non-linear manifold is perfectly captured by a non-linear method, then the PCA's reconstruction error would depend on how well a k-dimensional linear space can approximate the d-dimensional non-linear manifold.But since k > d, maybe the PCA can capture the manifold better? Or maybe not, because the manifold is non-linear.Wait, actually, if the manifold is non-linear, even a higher-dimensional linear space might not capture it well. For example, a sphere is a 2-dimensional manifold embedded in 3D space. A linear subspace of higher dimension (like 3D) can capture it exactly, but if the manifold is more complex, like a Swiss roll, a higher-dimensional linear space might not help much.Wait, but in our case, the non-linear method has zero reconstruction error, so it's perfect. So, the PCA's reconstruction error would be the difference between the data and its projection onto the PCA subspace.But since the data lies on a non-linear manifold, the PCA might not capture the true structure, leading to non-zero reconstruction error.But the question is, under what conditions is the PCA's reconstruction error minimized? That is, when is the PCA as good as it can be in approximating the non-linear manifold.I think this would happen when the non-linear manifold can be well-approximated by a linear subspace. For example, if the manifold is almost linear, then PCA can capture it well. But if the manifold is highly non-linear, then PCA might not do so well.But more formally, perhaps the reconstruction error is minimized when the non-linear manifold is close to a linear subspace. Alternatively, when the tangent spaces of the manifold are aligned with the PCA subspace.Alternatively, perhaps when the manifold can be locally approximated by the PCA subspace.Wait, but since the non-linear method has zero reconstruction error, it's not about local approximation but global.Alternatively, maybe the PCA's reconstruction error is minimized when the non-linear manifold is contained within the PCA subspace. But since the PCA subspace is linear and the manifold is non-linear, this is only possible if the manifold is itself linear, which contradicts the non-linear assumption.Wait, that can't be. If the manifold is non-linear, it can't be contained in a linear subspace unless it's a linear manifold, which it's not.So, perhaps the reconstruction error is minimized when the non-linear manifold is as close as possible to the PCA subspace. But how to formalize that.Alternatively, perhaps the reconstruction error is minimized when the PCA subspace is the best linear approximation to the non-linear manifold.But in that case, the reconstruction error would depend on how "non-linear" the manifold is. If the manifold is close to being linear, then the reconstruction error would be small.But the problem states that the non-linear method has zero reconstruction error, so it's perfect. So, the PCA's reconstruction error would be the difference between the data and the PCA subspace.Given that, the reconstruction error for PCA is the sum of squared distances from each data point to the PCA subspace. Since the non-linear method has zero error, the PCA's error is entirely due to the difference between the linear subspace and the non-linear manifold.So, to minimize PCA's reconstruction error, we need the non-linear manifold to be as close as possible to the PCA subspace. That is, the non-linear manifold should be such that projecting the data onto the PCA subspace results in minimal error.But since the manifold is non-linear, this might not be possible unless the manifold is approximately linear.Alternatively, perhaps when the non-linear manifold can be well-approximated by a linear subspace, then PCA's reconstruction error is minimized.But how to formalize this.Alternatively, perhaps the reconstruction error is minimized when the non-linear manifold is orthogonal to the PCA subspace. Wait, that might not make sense.Wait, maybe it's when the non-linear manifold lies within the PCA subspace. But as I thought earlier, that's only possible if the manifold is linear, which it's not.Alternatively, maybe when the non-linear manifold is contained within a higher-dimensional space, but the PCA subspace captures the main directions of variation.Wait, but since the non-linear method has zero error, the PCA's error is determined by how well the linear subspace can approximate the non-linear manifold.So, perhaps the reconstruction error is minimized when the non-linear manifold is such that the projection onto the PCA subspace is as accurate as possible.But without more specific information about the manifold, it's hard to say.Alternatively, perhaps the reconstruction error is minimized when the PCA subspace is the best linear approximation to the non-linear manifold, meaning that it captures the directions of maximum variance in the manifold.But since the manifold is non-linear, the variance might not be captured well by a linear subspace.Wait, maybe the reconstruction error is minimized when the non-linear manifold is isotropic, meaning that it doesn't have any particular direction of variance, so the PCA can capture it as well as possible.But I'm not sure.Alternatively, perhaps the reconstruction error is minimized when the non-linear manifold is such that the tangent spaces at various points are aligned with the PCA subspace.But that seems too vague.Alternatively, maybe the reconstruction error is minimized when the non-linear manifold is a linear subspace, but that contradicts the non-linear assumption.Wait, perhaps the key is that since the non-linear method has zero error, the PCA's reconstruction error is determined by the difference between the non-linear manifold and the linear subspace.So, the reconstruction error for PCA is the sum of squared distances from the data points to the PCA subspace. Since the data lies on the non-linear manifold, which is perfectly captured by the non-linear method, the PCA's reconstruction error is the sum of squared distances from the manifold to the PCA subspace.Therefore, to minimize PCA's reconstruction error, we need the non-linear manifold to be as close as possible to the PCA subspace.But how?Alternatively, perhaps the reconstruction error is minimized when the non-linear manifold is a linear subspace, but that's trivial and contradicts the non-linear assumption.Alternatively, perhaps when the non-linear manifold is contained within a higher-dimensional linear space, but that doesn't necessarily minimize the error.Wait, maybe the reconstruction error is minimized when the non-linear manifold is orthogonal to the PCA subspace. But that would maximize the error, not minimize it.Wait, no. If the manifold is orthogonal, the projection would be zero, but the reconstruction error would be large.Wait, perhaps the reconstruction error is minimized when the non-linear manifold is aligned with the PCA subspace as much as possible.But since the manifold is non-linear, it's not possible to align it completely.Alternatively, perhaps when the non-linear manifold is a small perturbation from a linear subspace, then PCA can capture it well, leading to minimal reconstruction error.But in that case, the non-linear method would have zero error, but PCA would have small error.So, in general, the reconstruction error for PCA is minimized when the non-linear manifold is as close as possible to a linear subspace, meaning that the non-linear structure is weak or the manifold is almost linear.But the problem states that the non-linear method has zero reconstruction error, so it's perfect. So, the PCA's reconstruction error is determined by how well the linear subspace can approximate the non-linear manifold.Therefore, the conditions under which PCA's reconstruction error is minimized would be when the non-linear manifold is as close as possible to a linear subspace. That is, when the non-linear structure is minimal or the manifold is approximately linear.But since the manifold is non-linear, this is a trade-off. The more non-linear the manifold, the higher the PCA's reconstruction error.Therefore, the reconstruction error is minimized when the non-linear manifold is closest to being linear, i.e., when the curvature of the manifold is minimal.Alternatively, when the manifold can be locally well-approximated by the PCA subspace.But since the non-linear method has zero error, it's a global property.Wait, perhaps another approach. The reconstruction error in PCA is the sum of the variances not captured by the first k principal components. So, if the data lies on a non-linear manifold, the variance not captured by the PCA subspace is the reconstruction error.But since the non-linear method captures the data perfectly, the PCA's reconstruction error is the difference between the data and the PCA subspace.Therefore, to minimize PCA's reconstruction error, we need the non-linear manifold to be as close as possible to the PCA subspace.But since the manifold is non-linear, this is only possible if the manifold is approximately linear.Therefore, the conditions are that the non-linear manifold is approximately linear, so that the PCA subspace can capture it well, minimizing the reconstruction error.But the problem states that the non-linear method has zero reconstruction error, so it's perfect. So, the PCA's error is due to the non-linearity of the manifold.Therefore, the reconstruction error is minimized when the non-linear manifold is as close as possible to a linear subspace, i.e., when the manifold is nearly linear.So, in summary, the reconstruction error using PCA is minimized when the non-linear manifold is approximately linear, meaning that the curvature or non-linearity of the manifold is minimal. In this case, the PCA can capture the structure well, leading to low reconstruction error.But if the manifold is highly non-linear, the PCA's reconstruction error would be higher.So, the conditions are that the non-linear manifold is close to being linear, so that the PCA subspace can approximate it well.But wait, the problem says that the intrinsic dimensionality is d < k < p. So, the PCA is using a higher-dimensional subspace than the intrinsic dimensionality.Does that help? Maybe, because a higher-dimensional linear space can better approximate a non-linear manifold. For example, a 2D non-linear manifold embedded in 3D space can be better approximated by a 3D linear space than a 2D linear space.But in our case, the non-linear method has zero error, so it's perfect. The PCA's reconstruction error is the difference between the data and the PCA subspace.But since k > d, the PCA subspace has higher dimensionality, which might allow it to capture more of the manifold's structure.Wait, but even with higher dimensionality, if the manifold is non-linear, the PCA might not capture it well. For example, a high-dimensional linear subspace can contain a non-linear manifold, but the projection onto the subspace might not align well with the manifold.Wait, no. If the non-linear manifold is embedded in a higher-dimensional space, and the PCA subspace is of higher dimensionality, then the projection of the data onto the PCA subspace might still not capture the non-linear structure.But since the non-linear method has zero error, the PCA's error is the difference between the data and the PCA subspace.So, perhaps the reconstruction error is minimized when the non-linear manifold is contained within the PCA subspace. But since the manifold is non-linear, it can't be contained in a linear subspace unless it's linear, which it's not.Therefore, the reconstruction error can't be zero, but it can be minimized when the manifold is as close as possible to the PCA subspace.But how?Alternatively, perhaps the reconstruction error is minimized when the PCA subspace is the best linear approximation to the non-linear manifold. That is, when the PCA subspace captures the main directions of variation in the manifold.But since the manifold is non-linear, the variance might not be captured well by a linear subspace.Alternatively, perhaps when the non-linear manifold is such that the tangent spaces at various points are aligned with the PCA subspace.But that's too vague.Alternatively, perhaps the reconstruction error is minimized when the non-linear manifold is a linear subspace, but that contradicts the non-linear assumption.Wait, maybe the key is that since the non-linear method has zero error, the PCA's reconstruction error is determined by the difference between the non-linear manifold and the linear subspace.Therefore, the reconstruction error is minimized when the non-linear manifold is as close as possible to the linear subspace.But since the manifold is non-linear, this is a trade-off. The more non-linear the manifold, the higher the PCA's reconstruction error.Therefore, the conditions are that the non-linear manifold is approximately linear, so that the PCA subspace can capture it well, minimizing the reconstruction error.But the problem states that the non-linear method has zero reconstruction error, so it's perfect. So, the PCA's error is due to the non-linearity of the manifold.Therefore, the reconstruction error is minimized when the non-linear manifold is as close as possible to a linear subspace, i.e., when the curvature or non-linearity of the manifold is minimal.So, in conclusion, the reconstruction error using PCA is minimized when the non-linear manifold is approximately linear, meaning that the curvature or non-linearity of the manifold is minimal. In this case, the PCA can capture the structure well, leading to low reconstruction error.However, if the manifold is highly non-linear, the PCA's reconstruction error would be higher.Additionally, since k > d, the PCA subspace has higher dimensionality, which might allow it to better approximate the non-linear manifold, but it's still limited by the linearity assumption.Therefore, the key condition is that the non-linear manifold is as close as possible to being linear, so that the PCA subspace can approximate it well.Final Answer1. The change in ( P_k ) due to small perturbations ( epsilon_i ) is approximately: [ Delta P_k approx frac{1}{S} left( sum_{i=1}^{k} epsilon_i - P_k sum_{j=1}^{p} epsilon_j right) ] where ( S = sum_{j=1}^{p} lambda_j ).2. The reconstruction error using PCA is minimized when the non-linear manifold is approximately linear, i.e., when the manifold's curvature or non-linearity is minimal. This allows the PCA subspace to better approximate the manifold, reducing the reconstruction error.The final answers are:1. (boxed{Delta P_k approx frac{1}{S} left( sum_{i=1}^{k} epsilon_i - P_k sum_{j=1}^{p} epsilon_j right)})2. The reconstruction error is minimized when the non-linear manifold is approximately linear, meaning the manifold's curvature is minimal.</think>

🧠:<think>Alright, so I have this problem about evaluating a software development project based on two metrics: Timely Delivery (T) and Code Quality (Q). The success of the project is modeled by the function S(T, Q) = a·ln(T + 1) + b·Q². There are two sub-problems to solve here.Starting with the first sub-problem: I need to find the critical points of S(T, Q) for a = 2 and b = 5, and determine their nature. The variables T and Q have ranges: T from 0 to 10 weeks, and Q from 0 to 1.Okay, critical points of a function of two variables are found by taking the partial derivatives with respect to each variable, setting them equal to zero, and solving the resulting system of equations. Then, we can use the second derivative test to determine if each critical point is a maximum, minimum, or saddle point.So, let me write down the function with a = 2 and b = 5:S(T, Q) = 2·ln(T + 1) + 5·Q².First, compute the partial derivatives.Partial derivative with respect to T:∂S/∂T = 2 / (T + 1).Partial derivative with respect to Q:∂S/∂Q = 10·Q.To find critical points, set both partial derivatives equal to zero.So, set ∂S/∂T = 0:2 / (T + 1) = 0.Hmm, solving for T. Let's see, 2 divided by something equals zero. The only way this can happen is if the denominator approaches infinity, but T is bounded between 0 and 10. So, 2 / (T + 1) is always positive for T in [0,10], since T + 1 is between 1 and 11. Therefore, 2 / (T + 1) can never be zero in this domain. That means there are no critical points where ∂S/∂T = 0.Wait, that seems odd. Maybe I made a mistake. Let me double-check.The partial derivative with respect to T is 2 / (T + 1). Since T is between 0 and 10, T + 1 is between 1 and 11, so 2 / (T + 1) is between 2/11 and 2, which is approximately 0.1818 to 2. So, it's always positive. Therefore, the partial derivative with respect to T never equals zero in the given domain. So, there are no critical points where both partial derivatives are zero.But wait, critical points can also occur on the boundaries of the domain. So, even if the partial derivatives don't equal zero inside the domain, we still need to check the boundaries for extrema.So, the function S(T, Q) is defined on a closed and bounded domain (rectangle in T-Q plane from (0,0) to (10,1)). Therefore, by the Extreme Value Theorem, the function must attain its maximum and minimum on this domain. These extrema can occur either at critical points inside the domain or on the boundary.But since we don't have any critical points inside the domain (because ∂S/∂T never equals zero), the extrema must occur on the boundary.So, to find the extrema, I need to evaluate S(T, Q) on the boundaries of the domain.The boundaries are:1. T = 0, Q varies from 0 to 1.2. T = 10, Q varies from 0 to 1.3. Q = 0, T varies from 0 to 10.4. Q = 1, T varies from 0 to 10.Additionally, the corners (0,0), (0,1), (10,0), (10,1) are also part of the boundaries.So, let's evaluate S on each of these boundaries.First, let's consider T = 0:S(0, Q) = 2·ln(0 + 1) + 5·Q² = 2·ln(1) + 5Q² = 0 + 5Q² = 5Q².This is a function of Q alone. To find extrema, take derivative with respect to Q:dS/dQ = 10Q. Setting to zero: 10Q = 0 => Q = 0.So, on T=0, the critical point is at Q=0. So, the point (0,0) is a critical point on this boundary.Similarly, on T=10:S(10, Q) = 2·ln(10 + 1) + 5Q² = 2·ln(11) + 5Q² ≈ 2·2.3979 + 5Q² ≈ 4.7958 + 5Q².Again, derivative with respect to Q is 10Q, set to zero: Q=0. So, (10,0) is another critical point on this boundary.Next, Q = 0:S(T, 0) = 2·ln(T + 1) + 5·0² = 2·ln(T + 1).Derivative with respect to T: 2 / (T + 1). Setting to zero: 2 / (T + 1) = 0. As before, no solution in T ∈ [0,10]. So, the extrema on Q=0 occur at the endpoints T=0 and T=10.Similarly, Q = 1:S(T, 1) = 2·ln(T + 1) + 5·1² = 2·ln(T + 1) + 5.Derivative with respect to T: 2 / (T + 1). Again, no critical points inside, so extrema at T=0 and T=10.Therefore, all the critical points on the boundaries are the four corners: (0,0), (0,1), (10,0), (10,1).Additionally, on the boundaries T=0 and T=10, we found critical points at Q=0, which are the same as the corners.So, now, to determine which of these points are maxima or minima, we need to evaluate S at each corner.Compute S at each corner:1. (0,0): S = 2·ln(1) + 5·0² = 0 + 0 = 0.2. (0,1): S = 2·ln(1) + 5·1² = 0 + 5 = 5.3. (10,0): S = 2·ln(11) + 5·0² ≈ 4.7958 + 0 ≈ 4.7958.4. (10,1): S = 2·ln(11) + 5·1² ≈ 4.7958 + 5 ≈ 9.7958.So, comparing these values:- Minimum at (0,0): 0.- Next, (10,0): ~4.7958.- Then, (0,1): 5.- Maximum at (10,1): ~9.7958.Therefore, the function S(T, Q) attains its minimum at (0,0) and maximum at (10,1). The other corners are in between.But wait, the problem says "find the critical points of the function S(T, Q)" and determine their nature.But earlier, we saw that there are no critical points inside the domain, only on the boundaries. So, the critical points are the four corners.But in the context of multivariable calculus, critical points are where the gradient is zero or undefined. In this case, the gradient is undefined nowhere in the domain, because ln(T + 1) is defined for T >= 0, and Q² is smooth everywhere. So, the only critical points would be where the gradient is zero, but as we saw, that doesn't happen inside the domain. So, actually, the function doesn't have any critical points in the interior, but the extrema occur on the boundaries.But in some contexts, people might refer to boundary points as critical points, but technically, critical points are where the gradient is zero or undefined. So, in this case, the function doesn't have any critical points in the interior, so the extrema are on the boundary, but they are not critical points in the traditional sense.Hmm, so maybe the answer is that there are no critical points in the domain, and the extrema occur at the corners.But let me double-check.Wait, in the boundaries, when we set T=0, we found a critical point at Q=0, which is (0,0). Similarly, on T=10, critical point at Q=0, which is (10,0). So, in a way, these are critical points on the boundary.But in the interior, there are no critical points because the partial derivatives never equal zero.So, perhaps the answer is that there are no critical points in the interior, but the extrema occur at the corners.But the question says "find the critical points of the function S(T, Q)", so perhaps they are considering the boundary points as critical points.Alternatively, maybe I need to consider that on the boundaries, the function reduces to a single variable function, and then critical points on those boundaries are considered.So, in that case, the critical points would be (0,0) and (10,0), as those are where the derivatives on the boundaries equal zero.But then, the other points (0,1) and (10,1) are just endpoints, not critical points.Wait, but in the context of the entire domain, the function S(T,Q) doesn't have any critical points except on the boundaries, where the partial derivatives can't be zero.So, perhaps the answer is that the only critical points are on the boundaries, specifically at (0,0) and (10,0), and these are minima.But when evaluating the function, (0,0) is the global minimum, and (10,1) is the global maximum. So, perhaps the critical points are (0,0) and (10,0), but (10,1) is just an endpoint maximum.Wait, but in the context of the entire domain, the function doesn't have any critical points except on the boundaries. So, the critical points are (0,0) and (10,0), which are both minima.But let me think again.In multivariable calculus, a critical point is a point where both partial derivatives are zero or where they don't exist. In this case, the partial derivatives exist everywhere in the domain, but they are never zero except on the boundaries.Wait, no. On the boundaries, the partial derivatives may not exist in the same way because we're restricting to the boundary.Actually, when considering the function on the entire domain, including the boundaries, the critical points would be where the gradient is zero or undefined. Since the gradient is defined everywhere, but only zero on the boundaries at (0,0) and (10,0). So, those are the only critical points.But wait, on the boundaries, the function is restricted, so the critical points on the boundaries would be where the derivative in the direction of the boundary is zero.But in our case, for T=0, we have S(0, Q) = 5Q², whose derivative is 10Q, which is zero at Q=0, so (0,0) is a critical point on that edge.Similarly, for T=10, S(10, Q) = 5Q² + constant, derivative 10Q, zero at Q=0, so (10,0) is a critical point on that edge.For Q=0, S(T, 0) = 2 ln(T + 1), derivative 2/(T + 1), which is never zero, so no critical points on Q=0 except at endpoints.Similarly, for Q=1, S(T,1) = 2 ln(T + 1) + 5, derivative 2/(T + 1), never zero, so no critical points on Q=1 except endpoints.Therefore, the only critical points are (0,0) and (10,0). The other points (0,1) and (10,1) are just endpoints, not critical points.So, now, to determine their nature: whether they are minima, maxima, or saddle points.Looking at (0,0): S(0,0) = 0. We saw that S is 0 there, and at (0,1) it's 5, at (10,0) it's ~4.7958, and at (10,1) ~9.7958. So, (0,0) is the global minimum.Similarly, (10,0): S(10,0) ≈4.7958. Comparing to nearby points, for example, moving a bit from (10,0) towards (10,1), S increases to 9.7958, and moving towards (9,0), S would be 2 ln(10) ≈4.6052, which is less than 4.7958. Wait, so moving from (10,0) towards (9,0), S decreases, and moving towards (10,1), S increases. So, at (10,0), S has a lower value in one direction and higher in another. Therefore, (10,0) is a saddle point.Wait, is that correct? Let me think.At (10,0), if we move along T decreasing, S decreases, and along Q increasing, S increases. So, in some directions, it's a minimum, in others, a maximum. Therefore, it's a saddle point.But wait, in the interior, saddle points are points where the function curves up in one direction and down in another. But here, (10,0) is on the boundary. So, in the interior, it's a saddle point, but on the boundary, it's a bit different.Wait, actually, in the context of the entire domain, (10,0) is a critical point where the function has a lower value in one direction (T decreasing) and higher in another (Q increasing). So, it's a saddle point.But let me confirm by using the second derivative test.For functions of two variables, the second derivative test involves computing the Hessian matrix:H = [f_TT f_TQ] [f_QT f_QQ]At a critical point, if the determinant D = f_TT * f_QQ - (f_TQ)^2 > 0 and f_TT > 0, then it's a local minimum; if D > 0 and f_TT < 0, it's a local maximum; if D < 0, it's a saddle point; if D = 0, the test is inconclusive.But in our case, the critical points are on the boundary, so the Hessian might not be directly applicable, but let's try.First, compute the second partial derivatives.f_TT = derivative of ∂S/∂T with respect to T: derivative of 2/(T + 1) is -2/(T + 1)^2.f_QQ = derivative of ∂S/∂Q with respect to Q: derivative of 10Q is 10.f_TQ = derivative of ∂S/∂T with respect to Q: 0, since ∂S/∂T doesn't depend on Q.Similarly, f_QT = 0.So, the Hessian is:[ -2/(T + 1)^2 0 ][ 0 10 ]Determinant D = (-2/(T + 1)^2)*10 - 0 = -20/(T + 1)^2.At (0,0): D = -20/(1)^2 = -20 < 0. So, it's a saddle point.Wait, but earlier, we saw that (0,0) is the global minimum. How come the second derivative test says it's a saddle point?Hmm, that seems contradictory. Maybe because (0,0) is on the boundary, the second derivative test isn't sufficient or applicable in the same way.Wait, actually, in the interior, if D < 0, it's a saddle point, but on the boundary, the behavior can be different. So, perhaps the second derivative test isn't the right tool here.Alternatively, maybe I should consider the behavior around the point within the domain.At (0,0): If we move in the T direction, S increases because ln(T + 1) increases as T increases. If we move in the Q direction, S increases because Q² increases as Q increases. So, (0,0) is a minimum.But according to the second derivative test, it's a saddle point. That seems conflicting.Wait, perhaps because the Hessian is negative definite in some directions and positive in others, but on the boundary, the movement is restricted.Alternatively, maybe the second derivative test isn't applicable at the boundary points because the function isn't defined beyond the boundary.So, perhaps, for the purposes of this problem, we can say that (0,0) is a minimum, and (10,0) is a saddle point, based on the behavior of the function.But let me think again.At (0,0): Any movement away from (0,0) in the domain (either increasing T or Q) increases S. So, it's a local minimum, and in fact, the global minimum.At (10,0): Moving in the T direction towards decreasing T, S decreases, and moving in the Q direction, S increases. So, it's a saddle point.Therefore, the critical points are (0,0) and (10,0). (0,0) is a local minimum, and (10,0) is a saddle point.But wait, in the interior, we don't have any critical points because the partial derivatives never equal zero. So, the only critical points are on the boundaries.So, to answer the first sub-problem: The critical points are (0,0) and (10,0). (0,0) is a local minimum, and (10,0) is a saddle point.But let me confirm with the second derivative test.At (0,0):f_TT = -2/(1)^2 = -2f_QQ = 10f_TQ = 0So, D = (-2)(10) - (0)^2 = -20 < 0. So, it's a saddle point.But in reality, (0,0) is the global minimum. So, perhaps the second derivative test is not applicable here because the point is on the boundary.Similarly, at (10,0):f_TT = -2/(11)^2 ≈ -2/121 ≈ -0.0165f_QQ = 10f_TQ = 0D = (-0.0165)(10) - 0 ≈ -0.165 < 0. So, again, saddle point.But in the case of (0,0), even though the second derivative test says saddle, the function actually has a minimum there. So, perhaps the test isn't reliable on the boundary.Therefore, perhaps the better approach is to analyze the behavior around the points.At (0,0): Any movement in positive T or Q increases S, so it's a minimum.At (10,0): Moving in negative T (towards T=9) decreases S, and moving in positive Q increases S. So, it's a saddle point.Therefore, the critical points are (0,0) and (10,0). (0,0) is a local minimum, and (10,0) is a saddle point.So, that's the answer for the first sub-problem.Now, moving on to the second sub-problem.The stakeholder requires that the sensitivity of the project success to changes in code quality (Q) should be twice as much as the sensitivity to changes in timely delivery (T). Determine the relationship between a and b that fulfills this requirement.Sensitivity in this context refers to the partial derivatives. So, the partial derivative of S with respect to Q should be twice the partial derivative with respect to T.So, mathematically:|∂S/∂Q| = 2 |∂S/∂T|But since both partial derivatives are positive in the domain (because T >=0 and Q >=0), we can drop the absolute values.So:∂S/∂Q = 2 ∂S/∂TCompute the partial derivatives:∂S/∂T = a / (T + 1)∂S/∂Q = 2b QSo, setting them in the required relationship:2b Q = 2 * (a / (T + 1))Simplify:2b Q = 2a / (T + 1)Divide both sides by 2:b Q = a / (T + 1)So, the relationship is b Q = a / (T + 1).But this is a relationship that must hold for all T and Q in the domain? Or is it a general relationship?Wait, the problem says "the sensitivity of the project success to changes in code quality should be twice as much as the sensitivity to changes in timely delivery." It doesn't specify for particular values of T and Q, so perhaps this should hold for all T and Q in the domain.But that seems impossible because the left side depends on Q and the right side depends on T. Unless Q and T are related in some way.Alternatively, maybe the stakeholder wants this condition to hold at a particular point, perhaps at the optimal point or at some specific operating point.But the problem doesn't specify, so perhaps we need to assume that this condition should hold for all T and Q, which would require that the functions of T and Q on each side are proportional.But let's see:b Q = a / (T + 1)This equation must hold for all T in [0,10] and Q in [0,1]. But unless both sides are constants, which they aren't, this can't hold for all T and Q.Therefore, perhaps the stakeholder wants this condition to hold at the optimal point, i.e., at the critical point where the project is most successful.But in our first sub-problem, the maximum occurs at (10,1). So, perhaps evaluating the condition at (10,1).At (10,1):∂S/∂Q = 2b * 1 = 2b∂S/∂T = a / (10 + 1) = a / 11So, setting 2b = 2*(a / 11)Simplify:2b = 2a / 11Divide both sides by 2:b = a / 11So, the relationship is b = a / 11.Alternatively, if the condition is to hold at the minimum point (0,0):At (0,0):∂S/∂Q = 0 (since Q=0)∂S/∂T = a / 1 = aSo, 0 = 2a, which implies a=0, which can't be because a is a weight.So, that doesn't make sense.Alternatively, maybe the condition is to hold at some specific point, but since it's not specified, perhaps the stakeholder wants it to hold in general, but that's impossible unless both sides are zero, which isn't useful.Alternatively, perhaps the stakeholder wants the ratio of the partial derivatives to be 2:1 on average or something, but that's more complicated.Alternatively, maybe the condition is that for any change in T and Q, the sensitivity to Q is twice that to T. But that would require the partial derivatives to satisfy ∂S/∂Q = 2 ∂S/∂T for all T and Q, which as we saw, is impossible unless both sides are zero, which isn't useful.Alternatively, perhaps the stakeholder wants the maximum sensitivity to Q to be twice the maximum sensitivity to T.Compute the maximum of ∂S/∂Q and ∂S/∂T.∂S/∂Q = 2b Q. Maximum when Q=1: 2b.∂S/∂T = a / (T + 1). Maximum when T=0: a.So, setting 2b = 2*(a). So, 2b = 2a => b = a.But that would make the maximum sensitivity of Q equal to the maximum sensitivity of T, not twice as much.Wait, the problem says sensitivity to Q should be twice as much as sensitivity to T. So, 2b = 2*(a). Wait, no.Wait, if the maximum sensitivity of Q is 2b, and the maximum sensitivity of T is a, then setting 2b = 2*(a) would mean b = a. But that's equal, not twice.Wait, perhaps the stakeholder wants the sensitivity to Q to be twice the sensitivity to T at all points, which would require 2b Q = 2*(a / (T + 1)), which simplifies to b Q = a / (T + 1). But as before, this can't hold for all T and Q unless both sides are constants, which they aren't.Alternatively, perhaps the stakeholder wants the ratio of the partial derivatives to be 2:1 on average over the domain. That would require integrating over the domain and setting the ratio of the integrals to 2:1.But that seems more complicated and not likely what the problem is asking.Alternatively, perhaps the stakeholder wants the partial derivatives to satisfy ∂S/∂Q = 2 ∂S/∂T at the point where the project is most successful, which is (10,1). As we saw earlier, that gives b = a / 11.Alternatively, perhaps the stakeholder wants this condition to hold at the point where the project is most sensitive to changes, which might be at the point where the partial derivatives are maximized.But the maximum sensitivity to Q is at Q=1, which is 2b, and the maximum sensitivity to T is at T=0, which is a. So, setting 2b = 2*(a) => b = a.But that would make the maximum sensitivity of Q equal to the maximum sensitivity of T, not twice as much.Wait, if we want the maximum sensitivity of Q to be twice the maximum sensitivity of T, then 2b = 2*(a) => b = a. Wait, no, that would make them equal. To make Q's maximum sensitivity twice that of T's, we need 2b = 2*(a) => b = a. Wait, no, that's still equal.Wait, no. If the maximum sensitivity of Q is 2b, and the maximum sensitivity of T is a, then to have 2b = 2*(a) => b = a. But that's equal, not twice.Wait, perhaps I'm getting confused.If the stakeholder wants the sensitivity to Q to be twice the sensitivity to T, then ∂S/∂Q = 2 ∂S/∂T.So, 2b Q = 2*(a / (T + 1)).Simplify: b Q = a / (T + 1).But unless Q and T are related, this can't hold for all T and Q. So, perhaps the stakeholder wants this condition to hold at a specific point, such as the optimal point.At the optimal point, which is (10,1), we have:∂S/∂Q = 2b*1 = 2b∂S/∂T = a / 11So, setting 2b = 2*(a / 11) => b = a / 11.Therefore, the relationship is b = a / 11.Alternatively, if the stakeholder wants this condition to hold at the point where the project is most sensitive to T, which is at T=0, Q=0, but there, ∂S/∂Q=0 and ∂S/∂T=a, so 0 = 2a, which is impossible.Alternatively, maybe the stakeholder wants this condition to hold on average over the domain.Compute the average sensitivity to Q and T.Average of ∂S/∂Q over the domain:(1 / (10*1)) ∫ (from T=0 to 10) ∫ (from Q=0 to 1) 2b Q dQ dT= (1/10) ∫0^10 [2b ∫0^1 Q dQ] dT= (1/10) ∫0^10 [2b*(1/2)] dT= (1/10) ∫0^10 b dT= (1/10)*b*10 = b.Similarly, average of ∂S/∂T over the domain:(1 / (10*1)) ∫0^10 ∫0^1 (a / (T + 1)) dQ dT= (1/10) ∫0^10 [a / (T + 1) * 1] dT= (1/10) * a * ∫0^10 1/(T + 1) dT= (1/10) * a * [ln(T + 1)] from 0 to 10= (1/10) * a * (ln(11) - ln(1)) = (1/10)*a*ln(11).So, the average sensitivity to Q is b, and the average sensitivity to T is (a ln(11))/10.The stakeholder wants the sensitivity to Q to be twice that to T on average:b = 2*(a ln(11)/10)=> b = (a ln(11))/5.So, the relationship is b = (a ln(11))/5.But the problem doesn't specify whether the condition should hold at a specific point or on average. Since it's a general requirement, perhaps the intended answer is to set the partial derivatives equal in the ratio 2:1 at the optimal point, which is (10,1), leading to b = a / 11.Alternatively, if it's on average, then b = (a ln(11))/5.But the problem says "the sensitivity of the project success to changes in code quality should be twice as much as the sensitivity to changes in timely delivery." It doesn't specify at a particular point or on average, so perhaps the intended interpretation is that the partial derivatives should satisfy ∂S/∂Q = 2 ∂S/∂T for all T and Q, which is impossible unless both sides are zero, which isn't useful. So, perhaps the next best interpretation is that this condition holds at the point where the project is most successful, which is (10,1), leading to b = a / 11.Alternatively, perhaps the stakeholder wants this condition to hold at the point where the project is most sensitive to T, which is at T=0, but there, ∂S/∂Q=0, which can't be twice anything.Alternatively, perhaps the stakeholder wants this condition to hold at the point where the project is most sensitive to Q, which is at Q=1, leading to 2b = 2*(a / (T + 1)). But unless T is fixed, this can't hold for all T.Alternatively, perhaps the stakeholder wants the ratio of the partial derivatives to be 2:1 at all points, which would require 2b Q = 2*(a / (T + 1)) => b Q = a / (T + 1). But unless Q and T are related, this can't hold for all T and Q.Therefore, the most plausible interpretation is that the condition holds at the optimal point, which is (10,1), leading to b = a / 11.Alternatively, if we consider the maximum sensitivities, the maximum sensitivity to Q is 2b (at Q=1), and the maximum sensitivity to T is a (at T=0). So, setting 2b = 2*(a) => b = a. But that would make the maximum sensitivity to Q equal to the maximum sensitivity to T, not twice as much.Wait, no. If we set 2b = 2*(a), that would mean b = a, but that's equal, not twice. To make the maximum sensitivity to Q twice the maximum sensitivity to T, we need 2b = 2*(a) => b = a. Wait, no, that's still equal. Wait, no, if we set 2b = 2*(a), then b = a. But that's equal, not twice.Wait, perhaps I'm overcomplicating. Let's go back.The problem says: "the sensitivity of the project success to changes in code quality should be twice as much as the sensitivity to changes in timely delivery."In terms of partial derivatives, this is ∂S/∂Q = 2 ∂S/∂T.So, 2b Q = 2*(a / (T + 1)).Simplify: b Q = a / (T + 1).But unless Q and T are related, this can't hold for all T and Q. So, perhaps the stakeholder wants this condition to hold at a specific point, such as the point where the project is most successful, which is (10,1). At that point, Q=1 and T=10, so:b*1 = a / (10 + 1) => b = a / 11.Therefore, the relationship is b = a / 11.Alternatively, if the stakeholder wants this condition to hold at the point where the project is most sensitive to T, which is at T=0, but there, Q=0, so ∂S/∂Q=0, which can't be twice anything.Alternatively, if the stakeholder wants this condition to hold at the point where the project is most sensitive to Q, which is at Q=1, but T can vary. So, at Q=1, for all T, 2b = 2*(a / (T + 1)) => b = a / (T + 1). But this would require T to be fixed, which isn't practical.Therefore, the most reasonable interpretation is that the condition holds at the optimal point (10,1), leading to b = a / 11.So, the relationship is b = a / 11.But let me check if this makes sense.If b = a / 11, then at (10,1):∂S/∂Q = 2b*1 = 2*(a / 11) = 2a / 11∂S/∂T = a / 11So, ∂S/∂Q = 2 ∂S/∂T, which satisfies the condition.Therefore, the relationship is b = a / 11.So, that's the answer for the second sub-problem.</think>