🧠:<think>Okay, so I have this problem about Alex, an ex-convict who's now into math and wants to model the success rate of rehabilitation programs. There are two parts to the problem. Let me tackle them one by one.Starting with the first part: Alex uses a logistic function to model the probability of successful reintegration, P(n), as a function of the number of therapy sessions, n. The function is given by:[ P(n) = frac{1}{1 + e^{-k(n - n_0)}} ]where k is 0.1 and n0 is 20. We need to find the number of sessions n required to achieve a 75% probability of success. So, P(n) = 0.75.Alright, let's write down what we know:- P(n) = 0.75- k = 0.1- n0 = 20We need to solve for n in the equation:[ 0.75 = frac{1}{1 + e^{-0.1(n - 20)}} ]Hmm, okay. Let me rearrange this equation step by step.First, take the reciprocal of both sides to get rid of the fraction:[ frac{1}{0.75} = 1 + e^{-0.1(n - 20)} ]Calculating 1/0.75, which is approximately 1.3333.So,[ 1.3333 = 1 + e^{-0.1(n - 20)} ]Subtract 1 from both sides:[ 0.3333 = e^{-0.1(n - 20)} ]Now, to solve for the exponent, take the natural logarithm of both sides:[ ln(0.3333) = -0.1(n - 20) ]Calculating ln(0.3333). I remember that ln(1/3) is approximately -1.0986.So,[ -1.0986 = -0.1(n - 20) ]Multiply both sides by -1 to make it positive:[ 1.0986 = 0.1(n - 20) ]Now, divide both sides by 0.1:[ 10.986 = n - 20 ]Add 20 to both sides:[ n = 10.986 + 20 ][ n = 30.986 ]Since the number of therapy sessions should be a whole number, we can round this up to 31 sessions. So, Alex needs approximately 31 sessions to reach a 75% success probability.Wait, let me double-check my calculations. Starting from P(n) = 0.75:0.75 = 1 / (1 + e^{-0.1(n - 20)})Yes, taking reciprocal:1 / 0.75 = 1 + e^{-0.1(n - 20)} => 1.3333 = 1 + e^{-0.1(n - 20)}Subtract 1: 0.3333 = e^{-0.1(n - 20)}Take ln: ln(0.3333) ≈ -1.0986 = -0.1(n - 20)Multiply both sides by -10: 10.986 = n - 20Add 20: n ≈ 30.986, so 31. That seems correct.Okay, moving on to part 2. Now, Alex incorporates a feedback loop into the model, considering community support S. The modified function is:[ P(n, S) = frac{1}{1 + e^{-k(n - n_0 + alpha S)}} ]Given that α = 0.05, S = 10, and the other parameters remain the same (k = 0.1, n0 = 20). We need to find the number of sessions n required to achieve the same 75% probability.So, similar to part 1, but now the exponent is adjusted by α*S.So, let's write down the equation:0.75 = 1 / (1 + e^{-0.1(n - 20 + 0.05*10)})First, calculate α*S: 0.05 * 10 = 0.5So, the exponent becomes:-0.1(n - 20 + 0.5) = -0.1(n - 19.5)Therefore, the equation is:0.75 = 1 / (1 + e^{-0.1(n - 19.5)})Again, let's solve for n.Take reciprocal:1 / 0.75 = 1 + e^{-0.1(n - 19.5)}1.3333 = 1 + e^{-0.1(n - 19.5)}Subtract 1:0.3333 = e^{-0.1(n - 19.5)}Take natural log:ln(0.3333) ≈ -1.0986 = -0.1(n - 19.5)Multiply both sides by -1:1.0986 = 0.1(n - 19.5)Divide by 0.1:10.986 = n - 19.5Add 19.5:n = 10.986 + 19.5 ≈ 30.486Again, rounding up, n ≈ 31 sessions.Wait, that's interesting. In part 1, n was approximately 31, and in part 2, it's also approximately 31. But let me check if I did that correctly.Wait, in part 2, the exponent is adjusted by α*S, which is 0.5. So, the term inside the exponent is (n - n0 + α*S) = (n - 20 + 0.5) = (n - 19.5). So, the exponent is -0.1(n - 19.5). So, when solving, we have:-1.0986 = -0.1(n - 19.5)Multiply both sides by -10:10.986 = n - 19.5So, n = 10.986 + 19.5 = 30.486, which is approximately 30.49. So, rounding up, 31 sessions.Wait, but in part 1, n was 30.986, which is about 31, and in part 2, n is 30.486, which is also about 31. So, actually, the number of sessions required is slightly less when community support is considered. Wait, but in this case, it's only slightly less, but still rounds up to 31.But let me think again. If S increases, does that mean the required n decreases? Because community support is positive, so it should make the exponent more positive, which would mean the probability increases faster, so you need fewer sessions to reach 75%.But in our calculation, n went from approximately 31 to approximately 30.5, which is a decrease, so that makes sense.But wait, in the calculation, the adjustment was adding α*S to (n - n0). So, the exponent becomes (n - n0 + α*S). So, effectively, it's like shifting the n0 to the left by α*S.So, n0 was 20, now it's 20 - 0.5 = 19.5. So, the midpoint is now at 19.5 instead of 20. So, to reach 75%, which is above the midpoint, you need fewer sessions because the curve is shifted to the left.So, yes, the number of sessions required is slightly less, but in both cases, it's about 31. So, depending on rounding, it might still be 31.But let me check if I did the calculations correctly.In part 2:0.75 = 1 / (1 + e^{-0.1(n - 20 + 0.05*10)})Which simplifies to:0.75 = 1 / (1 + e^{-0.1(n - 19.5)})Then, reciprocal:1.3333 = 1 + e^{-0.1(n - 19.5)}Subtract 1:0.3333 = e^{-0.1(n - 19.5)}Take ln:-1.0986 = -0.1(n - 19.5)Multiply by -10:10.986 = n - 19.5Add 19.5:n = 10.986 + 19.5 = 30.486So, 30.486, which is approximately 30.5. So, if we round to the nearest whole number, it's 31. But if we consider that 0.486 is less than 0.5, sometimes people round down, so 30. But in the context of therapy sessions, you can't have a fraction, so you'd need to complete 31 sessions to reach or exceed 75%.Wait, but let me check what P(30) and P(31) would be in part 2.Calculate P(30):P(30) = 1 / (1 + e^{-0.1(30 - 19.5)}) = 1 / (1 + e^{-0.1*10.5}) = 1 / (1 + e^{-1.05})e^{-1.05} ≈ 0.35So, P(30) ≈ 1 / (1 + 0.35) ≈ 1 / 1.35 ≈ 0.7407, which is about 74.07%, just below 75%.P(31):P(31) = 1 / (1 + e^{-0.1(31 - 19.5)}) = 1 / (1 + e^{-0.1*11.5}) = 1 / (1 + e^{-1.15})e^{-1.15} ≈ 0.316So, P(31) ≈ 1 / (1 + 0.316) ≈ 1 / 1.316 ≈ 0.76, which is above 75%.Therefore, to reach at least 75%, n needs to be 31.Similarly, in part 1, P(30) would be:P(30) = 1 / (1 + e^{-0.1(30 - 20)}) = 1 / (1 + e^{-1}) ≈ 1 / (1 + 0.3679) ≈ 0.721, which is below 75%.P(31) = 1 / (1 + e^{-0.1(31 - 20)}) = 1 / (1 + e^{-1.1}) ≈ 1 / (1 + 0.3329) ≈ 0.75, exactly 75%.Wait, actually, e^{-1.1} is approximately 0.3329, so 1 / (1 + 0.3329) ≈ 0.75. So, P(31) is exactly 75% in part 1.But in part 2, P(31) is about 76%, which is above 75%. So, the required n is still 31 in both cases, but in part 2, you reach 75% slightly earlier, but since you can't have a fraction of a session, you still need 31.Wait, but in part 2, n ≈30.486, so 30.486 sessions. So, if you do 30 sessions, you get about 74%, which is below 75%, and 31 sessions gets you to 76%, which is above. So, you still need 31 sessions.Therefore, the answer is 31 in both cases, but the calculation shows that with community support, you reach 75% slightly earlier in terms of the number of sessions, but since you can't have a fraction, it's still 31.Wait, but let me think again. The exact value in part 2 is 30.486, which is 30.486, so if we consider that, maybe in some contexts, you could say 30 sessions get you close, but not quite 75%, so you need 31.Alternatively, if the model allows for partial sessions, but in reality, you can't have partial sessions, so 31 is the answer.So, to summarize:1. Without community support, n ≈31.2. With community support, n ≈31.But wait, in part 2, the exact value is 30.486, which is less than 31, but since you can't have a fraction, you still need 31. So, the answer is 31 in both cases.But let me check if I did the calculations correctly.In part 1:n = 30.986 ≈31.In part 2:n = 30.486 ≈31.So, both require 31 sessions. Therefore, the number of sessions needed is 31 in both cases.Wait, but let me think about the impact of S. If S increases, the required n decreases. So, with S=10, n is slightly less than without S. So, in part 1, n was 31, and in part 2, it's 30.486, which is still 31 when rounded up.So, the answer is 31 in both cases.But wait, let me check if I made a mistake in the exponent in part 2.The function is P(n, S) = 1 / (1 + e^{-k(n - n0 + αS)}).So, the exponent is -k(n - n0 + αS) = -k(n - n0 + αS).So, in part 2, it's -0.1(n - 20 + 0.05*10) = -0.1(n - 20 + 0.5) = -0.1(n - 19.5).So, that's correct.So, solving for n, we get n ≈30.486, which is 30.486, so 31 when rounded up.Therefore, the answer is 31 in both cases.Wait, but let me think again. If S is positive, it shifts the curve to the left, meaning that for the same probability, you need fewer n. So, in part 2, n should be less than in part 1. But in both cases, it's 31. So, is that correct?Wait, in part 1, n was 30.986, which is almost 31, and in part 2, it's 30.486, which is still 31 when rounded up. So, the difference is minimal, but in reality, with S, you need slightly fewer sessions, but since you can't have a fraction, it's still 31.Alternatively, if we consider that 30.486 is closer to 30 than 31, maybe in some contexts, you could say 30 sessions, but since 30.486 is more than 30.48, which is more than 30.4, which is more than 30. So, depending on the rounding convention, it could be 30 or 31.But in the context of therapy sessions, you can't have a fraction, so you need to complete the next whole number. So, 31 sessions.Therefore, the answer is 31 in both cases.Wait, but let me check the exact value of P(30.486):P(n) = 1 / (1 + e^{-0.1(n - 19.5)}).At n=30.486:Exponent: -0.1*(30.486 - 19.5) = -0.1*(10.986) = -1.0986So, e^{-1.0986} ≈ 0.3333So, P(n) = 1 / (1 + 0.3333) ≈ 0.75, which is exactly 75%.So, at n=30.486, P(n)=0.75. So, if we could have 30.486 sessions, that's the exact point. But since we can't, we need to round up to 31.Therefore, the answer is 31 in both cases.So, to conclude:1. Without community support, n ≈31.2. With community support, n ≈31.But wait, in part 2, the exact n is 30.486, which is less than 31, but since we can't have a fraction, we still need 31.Therefore, the number of sessions required is 31 in both cases.Wait, but let me think again. If in part 2, n is 30.486, which is less than 31, but in reality, you can't have 0.486 of a session. So, you have to do 31 sessions to reach or exceed 75%.Therefore, the answer is 31 in both cases.So, final answers:1. 31 sessions.2. 31 sessions.But wait, let me check if I did the calculations correctly.In part 1:n = 30.986 ≈31.In part 2:n = 30.486 ≈31.Yes, both require 31 sessions.Therefore, the answers are both 31.</think>

🧠:<think>Alright, so I've got this problem about maximizing the Peace Impact Score (PIS) across three regions, A, B, and C. The PIS is given by the function ( P(x, y) = ln(x^2 + y^2 + 1) ). The journalist wants to maximize the total PIS across all three regions, with the constraint that the total number of successful peace initiatives, ( x_A + x_B + x_C ), is 50. Also, the public engagement ( y ) is constant at 10 for all regions. First, I need to figure out how to distribute the 50 successful initiatives among the three regions to maximize the total PIS. Since the PIS is a function of both ( x ) and ( y ), and ( y ) is fixed at 10, I can simplify the function for each region as ( P(x) = ln(x^2 + 10^2 + 1) = ln(x^2 + 101) ). So, the total PIS across all three regions would be ( P_A + P_B + P_C = ln(x_A^2 + 101) + ln(x_B^2 + 101) + ln(x_C^2 + 101) ). My goal is to maximize this sum given that ( x_A + x_B + x_C = 50 ).I remember that when trying to maximize or minimize a function subject to a constraint, Lagrange multipliers are a useful tool. So, maybe I should set up a Lagrangian function here. Let me recall how that works.The Lagrangian ( mathcal{L} ) would be the total PIS minus a multiplier times the constraint. So,[mathcal{L} = ln(x_A^2 + 101) + ln(x_B^2 + 101) + ln(x_C^2 + 101) - lambda (x_A + x_B + x_C - 50)]To find the maximum, I need to take the partial derivatives of ( mathcal{L} ) with respect to each ( x ) and ( lambda ), and set them equal to zero.Let's compute the partial derivative with respect to ( x_A ):[frac{partial mathcal{L}}{partial x_A} = frac{2x_A}{x_A^2 + 101} - lambda = 0]Similarly, for ( x_B ):[frac{partial mathcal{L}}{partial x_B} = frac{2x_B}{x_B^2 + 101} - lambda = 0]And for ( x_C ):[frac{partial mathcal{L}}{partial x_C} = frac{2x_C}{x_C^2 + 101} - lambda = 0]So, all three partial derivatives equal zero, which gives us the system of equations:1. ( frac{2x_A}{x_A^2 + 101} = lambda )2. ( frac{2x_B}{x_B^2 + 101} = lambda )3. ( frac{2x_C}{x_C^2 + 101} = lambda )And the constraint:4. ( x_A + x_B + x_C = 50 )From equations 1, 2, and 3, it seems that each ( frac{2x_i}{x_i^2 + 101} ) is equal to the same constant ( lambda ). This suggests that the expressions for each region are equal, which might imply that the optimal distribution is such that each ( x_i ) is equal? Or maybe not necessarily equal, but related in some way.Let me think. If all the ( frac{2x_i}{x_i^2 + 101} ) are equal, then perhaps each ( x_i ) is equal? Let me test that.Suppose ( x_A = x_B = x_C = x ). Then, since ( 3x = 50 ), ( x = 50/3 approx 16.6667 ). Let me check if this satisfies the condition.Compute ( frac{2x}{x^2 + 101} ) for ( x = 50/3 ):First, ( x^2 = (50/3)^2 = 2500/9 approx 277.7778 ). So, ( x^2 + 101 = 277.7778 + 101 = 378.7778 ).Then, ( 2x = 100/3 approx 33.3333 ). So, ( frac{33.3333}{378.7778} approx 0.088 ).So, each partial derivative would be approximately 0.088, which is equal, so that works. Therefore, the maximum occurs when each ( x_i = 50/3 ).Wait, but 50 divided by 3 is approximately 16.6667, which isn't an integer. Since the number of successful initiatives should be an integer, does this matter? The problem doesn't specify that ( x ) has to be an integer, so maybe it's okay to have fractional initiatives. Or perhaps, it's a continuous approximation.So, assuming ( x ) can be a real number, the optimal distribution is ( x_A = x_B = x_C = 50/3 approx 16.6667 ).But let me verify if this is indeed a maximum. Maybe I should check the second derivative or consider the nature of the function.The function ( P(x) = ln(x^2 + 101) ) is a concave function because its second derivative is negative. Let me compute the second derivative:First derivative: ( P'(x) = frac{2x}{x^2 + 101} )Second derivative: ( P''(x) = frac{2(x^2 + 101) - 2x(2x)}{(x^2 + 101)^2} = frac{2x^2 + 202 - 4x^2}{(x^2 + 101)^2} = frac{-2x^2 + 202}{(x^2 + 101)^2} )So, ( P''(x) = frac{-2x^2 + 202}{(x^2 + 101)^2} ). This is positive when ( -2x^2 + 202 > 0 ), which is when ( x^2 < 101 ), so ( |x| < sqrt{101} approx 10.05 ). Beyond that, the second derivative becomes negative.So, the function is concave for ( x > sqrt{101} approx 10.05 ) and convex for ( x < sqrt{101} ). Since our optimal ( x ) is approximately 16.6667, which is greater than 10.05, the function is concave there. Since the function is concave, the critical point we found is indeed a maximum. Therefore, distributing the initiatives equally among the three regions maximizes the total PIS.So, the values are ( x_A = x_B = x_C = 50/3 ).Wait, but let me think again. If the function is concave beyond a certain point, does that mean that distributing the initiatives equally is the optimal? Or is there a better distribution?Alternatively, maybe the function is symmetric, so equal distribution is optimal. Hmm.Alternatively, perhaps I can consider the derivative. Since each partial derivative is equal, it suggests that the marginal gain in PIS from each region is equal. So, if I take one initiative from one region and give it to another, the total PIS remains the same. Therefore, the maximum occurs when all regions have equal marginal gains, which is when all ( x_i ) are equal.So, I think my initial conclusion is correct.Therefore, the optimal distribution is ( x_A = x_B = x_C = 50/3 ).So, that answers the first part.Now, moving on to the second part. After maximizing the PIS, the journalist notices that the region with the highest PIS has a discrepancy in the recorded public engagement. Upon recalculating, the public engagement for that region is ( y + delta ), where ( delta ) is an unknown positive constant. I need to determine the derivative of the region's PIS with respect to ( delta ), and discuss under what conditions an increase in ( delta ) will lead to a decrease in the PIS.First, let's note that after maximizing, all regions have the same ( x ) value, which is 50/3, and the same ( y ) value, which is 10. So, all regions have the same PIS. Therefore, all regions have the same PIS, so technically, all regions have the highest PIS. Hmm, that might complicate things.Wait, but the problem says "the region with the highest PIS". But if all regions have the same PIS, then all are tied for the highest. So, perhaps the journalist is referring to one specific region, maybe the one where the discrepancy occurred. Alternatively, maybe the initial distribution wasn't equal, but that contradicts the first part.Wait, no, in the first part, we found that the optimal distribution is equal, so all regions have the same PIS. Therefore, all regions have the same PIS, so any region can be considered as the one with the highest PIS.But perhaps, for the sake of the problem, we can assume that one region's ( y ) is changed, while the others remain at 10. So, suppose region A's public engagement is now ( y + delta = 10 + delta ), while regions B and C remain at ( y = 10 ).Wait, but the problem says "the region with the highest PIS has a discrepancy". Since all regions have the same PIS, maybe the discrepancy is in one region, making its PIS higher or lower. But the problem says that upon recalculating, the public engagement is ( y + delta ), which is higher. So, perhaps the public engagement was underreported, and now it's corrected to a higher value.But the question is about the derivative of the region's PIS with respect to ( delta ). So, let's denote the region with the discrepancy as region A, for example. So, region A's PIS is now ( P_A = ln(x_A^2 + (10 + delta)^2 + 1) ), while regions B and C remain at ( P_B = ln(x_B^2 + 10^2 + 1) ) and ( P_C = ln(x_C^2 + 10^2 + 1) ).But wait, in the first part, we found that ( x_A = x_B = x_C = 50/3 ). So, all regions have the same ( x ). Therefore, region A's PIS becomes ( ln((50/3)^2 + (10 + delta)^2 + 1) ), while the others remain at ( ln((50/3)^2 + 10^2 + 1) ).But the problem says "the region's PIS", so perhaps we are to consider only the PIS of that region, not the total. So, we can treat the PIS of region A as a function of ( delta ), and find its derivative.So, let's define ( P_A(delta) = ln(x_A^2 + (10 + delta)^2 + 1) ). We need to find ( frac{dP_A}{ddelta} ).Compute the derivative:[frac{dP_A}{ddelta} = frac{2(10 + delta)}{x_A^2 + (10 + delta)^2 + 1}]Simplify the denominator:[x_A^2 + (10 + delta)^2 + 1 = x_A^2 + 100 + 20delta + delta^2 + 1 = x_A^2 + 101 + 20delta + delta^2]So, the derivative is:[frac{2(10 + delta)}{x_A^2 + 101 + 20delta + delta^2}]Now, we need to discuss under what conditions an increase in ( delta ) will lead to a decrease in the PIS. That is, when is ( frac{dP_A}{ddelta} < 0 )?But wait, the derivative is:[frac{2(10 + delta)}{x_A^2 + 101 + 20delta + delta^2}]Since ( delta ) is a positive constant, ( 10 + delta > 0 ), and the denominator is always positive because it's a sum of squares and positive constants. Therefore, the derivative ( frac{dP_A}{ddelta} ) is always positive. Wait, that can't be. If ( delta ) increases, the PIS should increase because the public engagement is higher, which would make the argument of the logarithm larger, hence the logarithm itself increases. So, the derivative should be positive, meaning that an increase in ( delta ) leads to an increase in PIS.But the problem asks under what conditions an increase in ( delta ) will lead to a decrease in the PIS. Hmm, that seems contradictory to what I just found.Wait, perhaps I made a mistake. Let me double-check.The PIS is ( ln(x^2 + y^2 + 1) ). If ( y ) increases, then ( y^2 ) increases, so the argument of the logarithm increases, hence the logarithm increases. Therefore, the derivative with respect to ( y ) should be positive.But in this case, we're taking the derivative with respect to ( delta ), which is added to ( y ). So, the derivative should indeed be positive.Wait, but the problem says "the region with the highest PIS has a discrepancy in the recorded public engagement. Upon recalculating, the public engagement for that region is ( y + delta ), where ( delta ) is an unknown positive constant." So, the public engagement was previously recorded as ( y ), but it's actually ( y + delta ). So, the PIS was previously calculated as ( ln(x^2 + y^2 + 1) ), but now it's ( ln(x^2 + (y + delta)^2 + 1) ). So, the PIS has increased because ( (y + delta)^2 > y^2 ). Therefore, the derivative of PIS with respect to ( delta ) is positive, meaning that an increase in ( delta ) leads to an increase in PIS.But the problem asks under what conditions an increase in ( delta ) will lead to a decrease in the PIS. That seems impossible based on the function. Unless... unless the region's PIS is being compared to the others, but the problem specifically says "the region's PIS", not the total PIS.Wait, perhaps I misinterpreted the problem. Maybe the journalist is considering the impact on the total PIS, not just the region's PIS. Let me read the problem again."After maximizing the Peace Impact Score, the journalist notices that the region with the highest PIS has a discrepancy in the recorded public engagement. Upon recalculating, the public engagement for that region is ( y + delta ), where ( delta ) is an unknown positive constant. Determine the derivative of the region's PIS with respect to ( delta ), and discuss under what conditions an increase in ( delta ) will lead to a decrease in the PIS."Hmm, it says "the region's PIS", so it's about the PIS of that specific region, not the total. So, as I computed, the derivative is positive, meaning that an increase in ( delta ) leads to an increase in PIS. Therefore, there are no conditions under which an increase in ( delta ) leads to a decrease in the PIS, because the derivative is always positive.But that seems odd. Maybe I need to consider the total PIS? Let me think.If the total PIS is considered, then increasing ( delta ) in one region would increase that region's PIS, but if the total initiatives are fixed, perhaps the distribution of ( x ) would change. But in the first part, we already maximized the total PIS under the constraint ( x_A + x_B + x_C = 50 ). So, if one region's ( y ) increases, the optimal distribution of ( x ) might change, potentially leading to a different total PIS.But the problem doesn't mention changing the distribution of ( x ); it just mentions that the public engagement for that region is ( y + delta ). So, perhaps the journalist is not re-optimizing the distribution of ( x ), but just recalculating the PIS with the corrected ( y ). Therefore, the total PIS would increase because one region's PIS increases, while the others remain the same.But the problem specifically asks about the derivative of the region's PIS with respect to ( delta ), not the total. So, as per the function, the derivative is positive, meaning an increase in ( delta ) leads to an increase in the region's PIS.Therefore, there are no conditions under which an increase in ( delta ) leads to a decrease in the PIS, because the derivative is always positive.Wait, but maybe I'm missing something. Perhaps the journalist is considering that increasing ( delta ) might cause the region's PIS to become too high, potentially overshadowing other regions or something. But the problem doesn't specify any such constraints or considerations. It just asks about the derivative and under what conditions an increase in ( delta ) leads to a decrease in the PIS.Given that, and based on the function, the derivative is always positive, so an increase in ( delta ) always leads to an increase in the PIS. Therefore, there are no conditions where an increase in ( delta ) decreases the PIS.But maybe I need to consider the second derivative or something else. Let me think.Wait, the function ( P(y) = ln(x^2 + y^2 + 1) ). The first derivative with respect to ( y ) is ( frac{2y}{x^2 + y^2 + 1} ), which is positive for ( y > 0 ). The second derivative with respect to ( y ) is ( frac{2(x^2 + y^2 + 1) - 4y^2}{(x^2 + y^2 + 1)^2} = frac{2x^2 + 2 - 2y^2}{(x^2 + y^2 + 1)^2} ). So, the second derivative is positive when ( x^2 + 1 > y^2 ), and negative otherwise.So, if ( y ) is such that ( y^2 < x^2 + 1 ), the function is convex, and if ( y^2 > x^2 + 1 ), it's concave.But in our case, ( x = 50/3 approx 16.6667 ), and ( y = 10 ). So, ( y^2 = 100 ), and ( x^2 + 1 approx (2500/9) + 1 approx 277.7778 + 1 = 278.7778 ). So, ( y^2 = 100 < 278.7778 ), so the second derivative is positive, meaning the function is convex in this region.But convexity doesn't directly affect whether the derivative is positive or negative. It just tells us about the curvature.So, regardless, the first derivative is positive, meaning that increasing ( delta ) (and hence ( y )) increases the PIS.Therefore, the derivative of the region's PIS with respect to ( delta ) is positive, and there are no conditions under which an increase in ( delta ) leads to a decrease in the PIS.Wait, but the problem says "discuss under what conditions an increase in ( delta ) will lead to a decrease in the PIS." So, perhaps I'm missing something.Alternatively, maybe the journalist is considering that if ( delta ) is too large, the region's PIS might start decreasing? But from the function, as ( y ) increases, ( P(y) ) increases without bound, because ( ln(y^2) ) grows as ( y ) grows. So, the PIS will keep increasing as ( y ) increases.Wait, no, actually, ( ln(y^2 + x^2 + 1) ) grows logarithmically, which is a slow growth, but it's still increasing as ( y ) increases.Therefore, I think the conclusion is that the derivative is always positive, so an increase in ( delta ) always leads to an increase in the region's PIS, and there are no conditions where it leads to a decrease.But the problem asks to discuss under what conditions an increase in ( delta ) will lead to a decrease in the PIS. Maybe I need to consider something else.Wait, perhaps the journalist is considering the impact on the total PIS, not just the region's PIS. If the journalist had to reallocate the initiatives, maybe increasing ( delta ) in one region could cause a decrease in another region's PIS, but that's not what the problem is asking.The problem specifically says "the region's PIS", so it's only about that region. Therefore, the derivative is positive, and an increase in ( delta ) leads to an increase in the region's PIS.Therefore, the conditions under which an increase in ( delta ) leads to a decrease in the PIS are non-existent, because the derivative is always positive.But maybe I need to express it differently. Perhaps, if ( delta ) were negative, but the problem states ( delta ) is a positive constant.Alternatively, maybe if ( delta ) is so large that it causes some other effect, but in the function, it's just a direct increase.So, in conclusion, the derivative of the region's PIS with respect to ( delta ) is positive, and there are no conditions under which an increase in ( delta ) leads to a decrease in the PIS.But wait, let me think again. Maybe I made a mistake in the derivative.Wait, the function is ( P(delta) = ln(x^2 + (10 + delta)^2 + 1) ). So, the derivative is:[frac{dP}{ddelta} = frac{2(10 + delta)}{x^2 + (10 + delta)^2 + 1}]Which is indeed positive for all ( delta > 0 ), since both numerator and denominator are positive.Therefore, the derivative is always positive, so an increase in ( delta ) always leads to an increase in the PIS.Therefore, there are no conditions where an increase in ( delta ) leads to a decrease in the PIS.So, to summarize:1. The optimal distribution is ( x_A = x_B = x_C = 50/3 ).2. The derivative of the region's PIS with respect to ( delta ) is positive, so an increase in ( delta ) always leads to an increase in the PIS, and there are no conditions where it leads to a decrease.But the problem specifically asks to discuss under what conditions an increase in ( delta ) will lead to a decrease in the PIS. Since the derivative is always positive, the answer is that there are no such conditions.Alternatively, maybe the problem is considering the total PIS, but as I thought earlier, if the journalist had to reallocate the initiatives, but the problem doesn't mention that. It just says the public engagement for that region is ( y + delta ), so the PIS for that region increases.Therefore, I think the answer is that the derivative is positive, and there are no conditions where an increase in ( delta ) leads to a decrease in the PIS.</think>

🧠:<think>Alright, so I've got these two sub-problems to solve related to a helicopter rescue mission. Let me take them one at a time and think through each step carefully.Starting with Sub-Problem 1: The helicopter had to adjust its flight path because of a storm. Initially, it was supposed to fly straight 100 km due north. But instead, it took a detour forming a triangle. The first leg was 60 km at an angle of 30 degrees east of due north, and the second leg was directly west. I need to find the total distance the helicopter traveled.Okay, so the original path was 100 km north. The new path is a triangle with two legs: 60 km at 30 degrees east of north, and then directly west. Hmm, so the first leg is 60 km, but not straight north—it's 30 degrees east of north. That means it has both a northward and eastward component. The second leg is directly west, so that should counteract the eastward component of the first leg, right? So the total displacement should still be 100 km north, but the path taken is longer.Wait, but the problem is asking for the total distance traveled, not the displacement. So I need to calculate the lengths of both legs and add them up. The first leg is 60 km, the second leg is directly west. I need to figure out how long the second leg is.Let me visualize this. The first leg is 60 km at 30 degrees east of north. So, breaking that into components: the northward component is 60 * cos(30°), and the eastward component is 60 * sin(30°). Let me calculate those.Cos(30°) is approximately 0.866, so 60 * 0.866 ≈ 51.96 km north. Sin(30°) is 0.5, so 60 * 0.5 = 30 km east.So after the first leg, the helicopter is 51.96 km north and 30 km east of the starting point. Then, the second leg is directly west. To get back to the original east-west line (since the original path was straight north), the helicopter needs to go west for 30 km to cancel out the eastward component.Therefore, the second leg is 30 km west. So the total distance traveled is the sum of the two legs: 60 km + 30 km = 90 km. Wait, but the original path was 100 km. So the detour is shorter? That doesn't make sense because detours usually make the distance longer. Did I do something wrong?Wait, no. Let me think again. The original path was 100 km north. The new path is 60 km at 30 degrees east of north, then 30 km west. So the total distance is 60 + 30 = 90 km, but the displacement is still 51.96 km north plus 30 km west, but wait, no—the second leg is west, so the eastward component is canceled. So the net displacement is 51.96 km north. But the original displacement was 100 km north. So the helicopter didn't reach the island yet? That can't be.Wait, maybe I misunderstood the problem. It says the helicopter had to take a detour forming a triangular path. So the initial point is the mainland, the island is 100 km north. The helicopter goes 60 km at 30 degrees east of north, then directly west to the island. So the second leg is not just 30 km west, but enough to reach the island.Wait, let me draw this. Starting point A, island at B, 100 km north. Helicopter goes from A to C, 60 km at 30 degrees east of north. Then from C to B, directly west. So the triangle is ACB, with AC = 60 km, angle at A is 30 degrees, and CB is directly west.So in triangle ACB, AC = 60 km, CB is west, AB = 100 km north. So we can use the Law of Cosines to find CB.Wait, but angle at A is 30 degrees. So in triangle ACB, angle at A is 30 degrees, side AC = 60 km, side AB = 100 km, and side CB is what we need to find.Law of Cosines: CB² = AC² + AB² - 2*AC*AB*cos(angle at A)So CB² = 60² + 100² - 2*60*100*cos(30°)Calculate that:60² = 3600100² = 100002*60*100 = 12000cos(30°) ≈ 0.866So CB² = 3600 + 10000 - 12000*0.866Calculate 12000*0.866 ≈ 10392So CB² ≈ 3600 + 10000 - 10392 = 13600 - 10392 = 3208Therefore, CB ≈ sqrt(3208) ≈ 56.64 kmSo the second leg is approximately 56.64 km west.Therefore, total distance traveled is 60 + 56.64 ≈ 116.64 km.Wait, that makes more sense because the detour would make the distance longer than 100 km.But let me double-check my approach. Alternatively, I could break down the first leg into components and then find the second leg.First leg: 60 km at 30 degrees east of north.North component: 60*cos(30) ≈ 51.96 kmEast component: 60*sin(30) = 30 kmSo from A to C: 51.96 km north, 30 km east.From C to B: needs to go west to reach the island, which is 100 km north of A.So the north component from C to B is 100 - 51.96 = 48.04 km north.But wait, the second leg is directly west, so it's purely westward. That means the helicopter is moving west while also needing to cover the remaining northward distance.Wait, that can't be right because if the second leg is directly west, it can't have a northward component. So perhaps my initial approach with the Law of Cosines is better.Alternatively, maybe the second leg is not directly west but directly towards the island, which would have both west and north components. But the problem says the second leg was directly west. So that means the helicopter is moving west, but also needs to cover the remaining northward distance. Hmm, that seems conflicting.Wait, perhaps the second leg is directly west, meaning it's moving west without any northward component. So the helicopter would have to adjust its position so that after moving west, it's directly above the island. But that would require that the westward movement cancels the eastward component, and the northward component is sufficient.Wait, but the first leg was 60 km at 30 degrees east of north, so the east component is 30 km. So to get back to the original meridian (directly north line), the helicopter needs to go 30 km west. But then, how much northward distance does it cover in that second leg?Wait, no. If the second leg is directly west, it's moving west without any northward movement. So the total northward distance covered is only the 51.96 km from the first leg. But the island is 100 km north, so the helicopter hasn't reached it yet. That doesn't make sense.Wait, maybe the second leg is not just west, but west and north? But the problem says directly west. Hmm, this is confusing.Let me re-examine the problem statement: "The second leg of the path was directly west." So it's a straight west path. So the helicopter goes 60 km at 30 degrees east of north, then turns and goes directly west until it reaches the island.But if it goes directly west, it can't cover any more northward distance. So unless the first leg already covered enough northward distance, the helicopter wouldn't reach the island. But the first leg only covered 51.96 km north, which is less than 100 km. So perhaps the second leg is not just west, but west and north? But the problem says directly west.Wait, maybe I need to consider that the second leg is west, but the helicopter is moving in such a way that it's both west and north? But no, directly west implies only westward movement.Wait, perhaps the second leg is not just a straight west line, but a vector that, when combined with the first leg, results in a total displacement of 100 km north. So the second leg would have both west and north components, but the problem says it's directly west. Hmm.Wait, maybe the second leg is directly west, meaning the helicopter is moving west while also ascending or something? No, that doesn't make sense in terms of flight path.Alternatively, perhaps the second leg is a straight line west that, when combined with the first leg, results in the total displacement of 100 km north. So using vector addition, the sum of the two legs should be 100 km north.So let me represent the first leg as a vector: 60 km at 30 degrees east of north. So in component form, that's (60*sin(30), 60*cos(30)) = (30, 51.96) km.The second leg is directly west, so its components are (-x, 0), where x is the distance west.The total displacement should be (0, 100) km because the island is 100 km north.So adding the two vectors:(30 - x, 51.96 + 0) = (0, 100)So 30 - x = 0 => x = 30 kmAnd 51.96 = 100? That's not possible. 51.96 ≠ 100.Wait, that's a contradiction. So this approach doesn't work because the northward component from the first leg is only 51.96 km, which is less than 100 km. So if the second leg is only west, the total northward displacement would still be 51.96 km, which is insufficient.Therefore, perhaps the second leg is not just west, but west and north? But the problem says directly west. Hmm.Wait, maybe the second leg is not a straight line west, but a vector that, when added to the first leg, results in a total displacement of 100 km north. So the second leg would have both west and north components, but the problem says it's directly west. So that seems conflicting.Alternatively, perhaps the second leg is a straight line west that, when combined with the first leg, results in the helicopter being at the island. So the second leg is a straight line from point C to point B, which is 100 km north of A.So point C is at (30, 51.96). Point B is at (0, 100). So the vector from C to B is (-30, 48.04). So the distance from C to B is sqrt(30² + 48.04²) ≈ sqrt(900 + 2307.36) ≈ sqrt(3207.36) ≈ 56.64 km.So the second leg is 56.64 km, but in what direction? The direction would be west and north, but the problem says directly west. So that contradicts.Wait, perhaps the problem means that the second leg is directly west, but the helicopter is moving in such a way that it's both west and north. But that would mean the second leg isn't purely west.Alternatively, maybe the problem is that the second leg is directly west, but the helicopter is moving west while also ascending or something? No, that doesn't make sense in terms of flight path.Wait, perhaps the problem is that the second leg is directly west, but the helicopter is moving west while also moving north? But that would mean the second leg has both west and north components, which contradicts the "directly west" part.Wait, maybe the problem is that the second leg is directly west, but the helicopter is moving west while also moving north? But that would mean the second leg has both west and north components, which contradicts the "directly west" part.Alternatively, perhaps the problem is that the second leg is directly west, but the helicopter is moving west while also moving north? But that would mean the second leg has both west and north components, which contradicts the "directly west" part.Wait, maybe I'm overcomplicating this. Let me try to think differently.If the first leg is 60 km at 30 degrees east of north, then the second leg is directly west. So the total displacement is 100 km north. So the sum of the two legs should be 100 km north.So the first leg has a north component of 60*cos(30) ≈ 51.96 km and an east component of 60*sin(30) = 30 km.The second leg is directly west, so it has a west component of x km and no north component.So the total displacement is (30 - x, 51.96) = (0, 100). Wait, that can't be because 51.96 ≠ 100.So that approach doesn't work. Therefore, perhaps the second leg is not just west, but west and north? But the problem says directly west.Wait, maybe the second leg is a straight line west that, when combined with the first leg, results in the helicopter being at the island. So the second leg is a straight line from point C to point B, which is 100 km north of A.So point C is at (30, 51.96). Point B is at (0, 100). So the vector from C to B is (-30, 48.04). So the distance from C to B is sqrt(30² + 48.04²) ≈ sqrt(900 + 2307.36) ≈ sqrt(3207.36) ≈ 56.64 km.So the second leg is 56.64 km, but in what direction? The direction would be west and north, but the problem says directly west. So that contradicts.Wait, perhaps the problem is that the second leg is directly west, but the helicopter is moving west while also moving north? But that would mean the second leg has both west and north components, which contradicts the "directly west" part.Alternatively, maybe the problem is that the second leg is directly west, but the helicopter is moving west while also moving north? But that would mean the second leg has both west and north components, which contradicts the "directly west" part.Wait, maybe the problem is that the second leg is directly west, but the helicopter is moving west while also moving north? But that would mean the second leg has both west and north components, which contradicts the "directly west" part.I think I'm stuck here. Let me try to approach it differently. Maybe the second leg is directly west, so the helicopter goes west for some distance, and then perhaps the total displacement is 100 km north. But that would require that the first leg's north component plus the second leg's north component equals 100 km. But the second leg is directly west, so it has no north component. Therefore, the first leg's north component must be 100 km. But the first leg is only 60 km at 30 degrees, which gives a north component of 51.96 km, which is less than 100 km. Therefore, this approach doesn't work.Wait, maybe the second leg is not just west, but west and north? But the problem says directly west. Hmm.Alternatively, perhaps the second leg is a straight line west that, when combined with the first leg, results in the helicopter being at the island. So the second leg is a straight line from point C to point B, which is 100 km north of A.So point C is at (30, 51.96). Point B is at (0, 100). So the vector from C to B is (-30, 48.04). So the distance from C to B is sqrt(30² + 48.04²) ≈ sqrt(900 + 2307.36) ≈ sqrt(3207.36) ≈ 56.64 km.So the second leg is 56.64 km, but in what direction? The direction would be west and north, but the problem says directly west. So that contradicts.Wait, maybe the problem is that the second leg is directly west, but the helicopter is moving west while also moving north? But that would mean the second leg has both west and north components, which contradicts the "directly west" part.I think I need to accept that the second leg is directly west, and the total displacement is 100 km north. Therefore, the first leg's north component plus the second leg's north component must equal 100 km. But the second leg is directly west, so it has no north component. Therefore, the first leg's north component must be 100 km. But the first leg is only 60 km at 30 degrees, which gives a north component of 51.96 km, which is less than 100 km. Therefore, this approach doesn't work.Wait, maybe the second leg is not just west, but west and north? But the problem says directly west. Hmm.Alternatively, perhaps the second leg is a straight line west that, when combined with the first leg, results in the helicopter being at the island. So the second leg is a straight line from point C to point B, which is 100 km north of A.So point C is at (30, 51.96). Point B is at (0, 100). So the vector from C to B is (-30, 48.04). So the distance from C to B is sqrt(30² + 48.04²) ≈ sqrt(900 + 2307.36) ≈ sqrt(3207.36) ≈ 56.64 km.So the second leg is 56.64 km, but in what direction? The direction would be west and north, but the problem says directly west. So that contradicts.Wait, maybe the problem is that the second leg is directly west, but the helicopter is moving west while also moving north? But that would mean the second leg has both west and north components, which contradicts the "directly west" part.I think I need to conclude that the second leg is 56.64 km west, but that would mean the helicopter is moving west and north, which contradicts the problem statement. Therefore, perhaps the problem is intended to have the second leg as 30 km west, and the total distance is 60 + 30 = 90 km, but that would mean the helicopter didn't reach the island because the north component is only 51.96 km. Therefore, perhaps the problem is intended to have the second leg as 56.64 km west, but that contradicts the "directly west" part.Wait, maybe the second leg is directly west, but the helicopter is moving west while also moving north? But that would mean the second leg has both west and north components, which contradicts the "directly west" part.I think I need to go back to the Law of Cosines approach. The triangle is ACB, with AC = 60 km, AB = 100 km, angle at A = 30 degrees. Therefore, CB can be found using the Law of Cosines.CB² = AC² + AB² - 2*AC*AB*cos(angle at A)CB² = 60² + 100² - 2*60*100*cos(30°)CB² = 3600 + 10000 - 12000*0.866CB² = 13600 - 10392 = 3208CB ≈ sqrt(3208) ≈ 56.64 kmTherefore, the second leg is approximately 56.64 km west. So the total distance traveled is 60 + 56.64 ≈ 116.64 km.But the problem says the second leg is directly west, so the direction is west, but the distance is 56.64 km. Therefore, the total distance is 60 + 56.64 ≈ 116.64 km.So I think that's the answer. The total distance traveled is approximately 116.64 km.Now, moving on to Sub-Problem 2: The survivor was hoisted up to the helicopter at a constant speed of 2 m/s. The height from the survivor's position to the helicopter cabin was 60 meters, and the helicopter was moving horizontally at a constant speed of 10 m/s. Determine the resultant velocity of the survivor as they were being hoisted up.Okay, so the survivor is being hoisted up at 2 m/s vertically, while the helicopter is moving horizontally at 10 m/s. So the survivor's velocity is a combination of vertical and horizontal components.Since these are perpendicular components, the resultant velocity can be found using the Pythagorean theorem.So the horizontal component is 10 m/s, and the vertical component is 2 m/s.Resultant velocity, V = sqrt((10)^2 + (2)^2) = sqrt(100 + 4) = sqrt(104) ≈ 10.198 m/s.But let me think if there's more to it. The problem mentions the height is 60 meters, but since we're dealing with velocity, which is a rate, the height might not be directly relevant unless we're considering the time taken, but the problem doesn't ask for time, just the resultant velocity.So yes, the resultant velocity is the vector sum of the horizontal and vertical components, which is sqrt(10^2 + 2^2) ≈ 10.198 m/s.But let me double-check. The survivor is moving up at 2 m/s and the helicopter is moving forward at 10 m/s. So the survivor's velocity relative to the ground is the vector sum of these two. So yes, the magnitude is sqrt(10^2 + 2^2).Alternatively, if we consider the survivor's velocity relative to the helicopter, it's just 2 m/s upwards, but relative to the ground, it's 10 m/s horizontally and 2 m/s vertically.Therefore, the resultant velocity is approximately 10.198 m/s.But let me calculate it more precisely. sqrt(104) is approximately 10.198039 m/s, which we can round to 10.2 m/s.So, to summarize:Sub-Problem 1: Total distance traveled is approximately 116.64 km.Sub-Problem 2: Resultant velocity is approximately 10.2 m/s.Wait, but for Sub-Problem 1, I think I should present it as 116.64 km, but maybe the problem expects an exact value. Let me see:CB² = 60² + 100² - 2*60*100*cos(30°)cos(30°) = √3/2 ≈ 0.8660254So CB² = 3600 + 10000 - 12000*(√3/2)= 13600 - 6000√3So CB = sqrt(13600 - 6000√3)But that's an exact expression, but it's probably better to compute it numerically.sqrt(13600 - 6000*1.73205) ≈ sqrt(13600 - 10392.3) ≈ sqrt(3207.7) ≈ 56.64 kmSo total distance is 60 + 56.64 ≈ 116.64 km.Alternatively, if we keep it exact, it's 60 + sqrt(13600 - 6000√3). But I think the numerical value is acceptable.For Sub-Problem 2, the exact value is sqrt(104) m/s, which is 2*sqrt(26) m/s, but again, the decimal is more practical.So I think I've got both problems solved.</think>

🧠:<think>Okay, so I have this problem about a podcast host analyzing gold mining operations. There are two parts. Let me try to tackle them one by one.Starting with the first part: The host models revenue R as a function of gold extracted G with the equation R(G) = 5G² - 3G + 2. I need to find the critical points and classify them.Hmm, critical points. I remember from calculus that critical points occur where the derivative is zero or undefined. Since this is a quadratic function, its derivative should be a linear function, which is defined everywhere, so critical points will just be where the derivative is zero.Let me compute the derivative of R(G). The derivative of R with respect to G is R’(G) = d/dG [5G² - 3G + 2]. So, derivative of 5G² is 10G, derivative of -3G is -3, and the derivative of 2 is 0. So R’(G) = 10G - 3.To find critical points, set R’(G) = 0:10G - 3 = 010G = 3G = 3/10G = 0.3So the critical point is at G = 0.3 tons.Now, I need to classify this critical point as a local maximum, local minimum, or neither. Since R(G) is a quadratic function, and the coefficient of G² is positive (5), the parabola opens upwards. That means the critical point is a local minimum.Wait, let me think again. If the parabola opens upwards, the vertex is the minimum point. So yes, G = 0.3 is a local minimum.But just to be thorough, maybe I should use the second derivative test.Compute the second derivative of R(G). The first derivative was 10G - 3, so the second derivative is 10, which is positive. Since the second derivative is positive, the function is concave up at that point, confirming it's a local minimum.So, first part done. The critical point is at G = 0.3 tons, and it's a local minimum.Moving on to the second part: The host now considers the cost C(G) = (1/3)G³ - G + 4. We need to find the amount of gold G that should be extracted to maximize the profit P(G) = R(G) - C(G). Then determine the maximum profit.Alright, so profit is revenue minus cost. Let me write that out:P(G) = R(G) - C(G) = [5G² - 3G + 2] - [(1/3)G³ - G + 4]Let me simplify this expression step by step.First, distribute the negative sign to each term in C(G):= 5G² - 3G + 2 - (1/3)G³ + G - 4Now, combine like terms.Let's write the terms in order of descending powers:- (1/3)G³ + 5G² + (-3G + G) + (2 - 4)Simplify each part:- (1/3)G³ + 5G² - 2G - 2So, P(G) = - (1/3)G³ + 5G² - 2G - 2Now, to find the maximum profit, we need to find the critical points of P(G). Again, critical points occur where the derivative is zero or undefined. Since P(G) is a cubic polynomial, its derivative will be a quadratic function, which is defined everywhere, so critical points are where the derivative is zero.Compute the first derivative P’(G):P’(G) = d/dG [ - (1/3)G³ + 5G² - 2G - 2 ]Derivative of - (1/3)G³ is -G², derivative of 5G² is 10G, derivative of -2G is -2, and derivative of -2 is 0.So, P’(G) = -G² + 10G - 2Set this equal to zero to find critical points:- G² + 10G - 2 = 0Let me rewrite this equation to make it a bit easier:G² - 10G + 2 = 0Wait, I multiplied both sides by -1 to make the coefficient of G² positive, which is a common step to make it easier to solve.So, G² - 10G + 2 = 0Now, this is a quadratic equation. Let me use the quadratic formula to solve for G.Quadratic formula: G = [10 ± sqrt( (-10)^2 - 4*1*2 )]/(2*1)Compute discriminant D:D = (-10)^2 - 4*1*2 = 100 - 8 = 92So, sqrt(D) = sqrt(92). Let me simplify sqrt(92). 92 is 4*23, so sqrt(4*23) = 2*sqrt(23). So sqrt(92) = 2*sqrt(23).Therefore, solutions are:G = [10 ± 2*sqrt(23)] / 2Simplify numerator:= [10 + 2*sqrt(23)] / 2 and [10 - 2*sqrt(23)] / 2Factor out 2 in numerator:= 2[5 + sqrt(23)] / 2 and 2[5 - sqrt(23)] / 2Cancel 2:G = 5 + sqrt(23) and G = 5 - sqrt(23)Compute approximate values to understand the critical points.sqrt(23) is approximately 4.796.So, G ≈ 5 + 4.796 ≈ 9.796And G ≈ 5 - 4.796 ≈ 0.204So, critical points at approximately G ≈ 0.204 and G ≈ 9.796.Now, since we are dealing with a cubic function P(G) = - (1/3)G³ + 5G² - 2G - 2, which has a negative leading coefficient, the cubic tends to negative infinity as G approaches positive infinity and positive infinity as G approaches negative infinity. Therefore, the function will have a local maximum and a local minimum.Given that, the critical point at G ≈ 0.204 is likely a local maximum, and G ≈ 9.796 is a local minimum. But let me confirm using the second derivative test.Compute the second derivative P''(G):P’(G) = -G² + 10G - 2So, P''(G) = -2G + 10Now, evaluate P''(G) at each critical point.First, at G = 5 - sqrt(23) ≈ 0.204:P''(0.204) = -2*(0.204) + 10 ≈ -0.408 + 10 ≈ 9.592Since this is positive, the function is concave up at this point, meaning it's a local minimum? Wait, hold on. Wait, no, wait.Wait, if the second derivative is positive, that means the function is concave up, which implies a local minimum. But earlier, I thought G ≈ 0.204 was a local maximum. Hmm, seems conflicting.Wait, perhaps I made a mistake in the earlier assumption.Wait, let's think again. The cubic function P(G) has a negative leading coefficient, so as G increases, the function tends to negative infinity. So, the function will rise to a local maximum, then fall to a local minimum, and then continue to negative infinity.Therefore, the critical point at G ≈ 0.204 is a local maximum, and G ≈ 9.796 is a local minimum.But according to the second derivative test, at G ≈ 0.204, P''(G) is positive, which suggests concave up, hence a local minimum. That contradicts the earlier reasoning.Wait, perhaps I messed up the second derivative.Wait, let me compute P''(G) again.P’(G) = -G² + 10G - 2So, P''(G) is the derivative of that, which is -2G + 10. Correct.So, at G ≈ 0.204, P''(G) ≈ -2*(0.204) + 10 ≈ -0.408 + 10 ≈ 9.592, which is positive. So, concave up, so local minimum.But according to the shape of the cubic, it should be a local maximum. Hmm, maybe my initial assumption about the shape was wrong.Wait, let me plot the function or think about the behavior.Wait, as G approaches negative infinity, since the leading term is - (1/3)G³, which for G negative, becomes positive infinity. So, as G approaches negative infinity, P(G) approaches positive infinity. As G approaches positive infinity, P(G) approaches negative infinity.Therefore, the function will come from positive infinity, decrease to a local minimum, then increase to a local maximum, and then decrease to negative infinity.Wait, that seems conflicting with my earlier thought.Wait, no, actually, the leading term is negative, so as G increases, the function tends to negative infinity. So, the function will start from positive infinity when G is negative, decrease, reach a local minimum, then increase to a local maximum, and then decrease again to negative infinity.Therefore, the critical points are: first a local minimum at G ≈ 0.204, then a local maximum at G ≈ 9.796.Wait, that makes sense because the second derivative at G ≈ 0.204 is positive (local minimum), and at G ≈ 9.796, let's compute the second derivative:P''(9.796) = -2*(9.796) + 10 ≈ -19.592 + 10 ≈ -9.592, which is negative, so concave down, hence local maximum.Ah, okay, so my initial thought was reversed. The first critical point is a local minimum, and the second is a local maximum.But wait, the question is about maximizing profit. So, the maximum profit occurs at G ≈ 9.796 tons.But let me confirm.Wait, so the profit function P(G) has a local minimum at G ≈ 0.204 and a local maximum at G ≈ 9.796. So, the maximum profit is at G ≈ 9.796 tons.But wait, let me think about the domain of G. G represents tons of gold extracted, so it must be a positive value. So, G must be greater than zero.But is there any upper limit on G? The problem doesn't specify, so theoretically, G can be any positive number. However, in reality, there might be constraints, but since it's not given, we proceed with the mathematical result.Therefore, the maximum profit occurs at G ≈ 9.796 tons.But let me compute the exact value instead of the approximate.We had G = 5 + sqrt(23) and G = 5 - sqrt(23). Since sqrt(23) is approximately 4.796, G = 5 + sqrt(23) is approximately 9.796, and G = 5 - sqrt(23) is approximately 0.204.So, the exact critical points are G = 5 ± sqrt(23). Since 5 - sqrt(23) is approximately 0.204, which is a local minimum, and 5 + sqrt(23) is approximately 9.796, which is a local maximum.Therefore, the amount of gold G that should be extracted to maximize profit is G = 5 + sqrt(23) tons.But let me compute the exact value of the maximum profit.So, plug G = 5 + sqrt(23) into P(G):P(G) = - (1/3)G³ + 5G² - 2G - 2This might get a bit messy, but let's try.First, let me denote sqrt(23) as s for simplicity.So, G = 5 + s, where s = sqrt(23)Compute G³:G³ = (5 + s)³ = 125 + 75s + 15s² + s³But s² = 23, and s³ = s*s² = s*23 = 23sSo, G³ = 125 + 75s + 15*23 + 23s = 125 + 75s + 345 + 23s = (125 + 345) + (75s + 23s) = 470 + 98sSimilarly, G² = (5 + s)² = 25 + 10s + s² = 25 + 10s + 23 = 48 + 10sNow, plug into P(G):P(G) = - (1/3)G³ + 5G² - 2G - 2= - (1/3)(470 + 98s) + 5*(48 + 10s) - 2*(5 + s) - 2Compute each term:- (1/3)(470 + 98s) = -470/3 - (98/3)s ≈ -156.6667 - 32.6667sBut let's keep it exact:= -470/3 - (98/3)s5*(48 + 10s) = 240 + 50s-2*(5 + s) = -10 - 2s-2 remains as is.So, combine all terms:= (-470/3 - 98/3 s) + (240 + 50s) + (-10 - 2s) + (-2)Convert all terms to thirds to combine:-470/3 - 98/3 s + 240*(3/3) + 50s*(3/3) -10*(3/3) -2s*(3/3) -2*(3/3)Wait, maybe a better approach is to convert all constants and coefficients to thirds.Alternatively, let's compute each part step by step.First, constants:-470/3 + 240 -10 -2Convert 240, -10, -2 to thirds:240 = 720/3-10 = -30/3-2 = -6/3So, constants:-470/3 + 720/3 -30/3 -6/3 = (-470 + 720 -30 -6)/3 = (720 - 470 -36)/3 = (250 -36)/3 = 214/3 ≈ 71.3333Now, coefficients of s:-98/3 s + 50s -2sConvert 50s and -2s to thirds:50s = 150/3 s-2s = -6/3 sSo, total s terms:(-98/3 + 150/3 -6/3)s = (46/3)s ≈ 15.3333sTherefore, P(G) = 214/3 + (46/3)sBut s = sqrt(23), so:P(G) = (214 + 46 sqrt(23))/3We can factor numerator:= (214 + 46 sqrt(23))/3Alternatively, factor 2:= 2*(107 + 23 sqrt(23))/3But maybe that's not necessary. So, the exact maximum profit is (214 + 46 sqrt(23))/3 million dollars.Alternatively, we can write it as:P(G) = (214 + 46√23)/3Which is approximately:Compute sqrt(23) ≈ 4.79646*4.796 ≈ 46*4 + 46*0.796 ≈ 184 + 36.616 ≈ 220.616214 + 220.616 ≈ 434.616Divide by 3: ≈ 144.872 million dollars.So, approximately 144.872 million dollars.But let me check my calculations because this seems a bit high.Wait, let me re-express P(G) at G = 5 + sqrt(23):P(G) = - (1/3)G³ + 5G² - 2G - 2We had G³ = 470 + 98s, G² = 48 + 10s, where s = sqrt(23).So:- (1/3)(470 + 98s) = -470/3 - 98s/35*(48 + 10s) = 240 + 50s-2*(5 + s) = -10 - 2s-2 remains.So, combining constants:-470/3 + 240 -10 -2 = (-470/3) + 228Convert 228 to thirds: 228 = 684/3So, constants: (-470 + 684)/3 = 214/3Coefficients of s:-98/3 s + 50s -2s = (-98/3 + 48/3)s = (-50/3)sWait, hold on, 50s -2s is 48s, which is 48/3 s in thirds.So, total s terms:-98/3 s + 48/3 s = (-50/3)sWait, that contradicts my earlier calculation. So, seems I made a mistake earlier.So, correct calculation:Coefficients of s:From - (1/3)(470 + 98s): -98/3 sFrom 5*(48 + 10s): +50sFrom -2*(5 + s): -2sSo, total s terms:-98/3 s + 50s - 2s = (-98/3 + 48/3)s = (-50/3)sWait, 50s -2s = 48s, which is 48/3 s. So, -98/3 + 48/3 = (-50)/3.So, s terms: (-50/3)sTherefore, total P(G) = 214/3 - (50/3)sWhich is (214 - 50s)/3Since s = sqrt(23), P(G) = (214 - 50 sqrt(23))/3Wait, that's different from what I had earlier. So, I must have messed up the signs.Wait, let's go back.Original expression:P(G) = - (1/3)G³ + 5G² - 2G - 2We computed G³ = 470 + 98s, G² = 48 + 10sSo,- (1/3)G³ = - (470 + 98s)/35G² = 5*(48 + 10s) = 240 + 50s-2G = -2*(5 + s) = -10 -2s-2 remains.So, constants:-470/3 + 240 -10 -2= (-470/3) + (240 -10 -2)= (-470/3) + 228Convert 228 to thirds: 228 = 684/3So, constants: (-470 + 684)/3 = 214/3s terms:-98s/3 + 50s -2s= (-98/3)s + 48sConvert 48s to thirds: 48s = 144/3 sSo, total s terms:(-98/3 + 144/3)s = (46/3)sTherefore, P(G) = 214/3 + (46/3)sWhich is (214 + 46 sqrt(23))/3Wait, so my initial calculation was correct, but when I thought I made a mistake, I messed up the signs.So, P(G) = (214 + 46 sqrt(23))/3Which is approximately:46*4.796 ≈ 220.616214 + 220.616 ≈ 434.616Divide by 3: ≈ 144.872 million dollars.So, approximately 144.872 million dollars.But let me verify once more because I initially thought the s terms would be negative, but it's positive.Wait, in the expression:P(G) = - (1/3)G³ + 5G² - 2G - 2So, when we plug in G = 5 + sqrt(23), which is a large G, the negative term - (1/3)G³ will dominate, but since we are at a local maximum, the profit is positive?Wait, but 144 million seems high. Let me compute P(G) numerically for G ≈ 9.796.Compute P(G) = - (1/3)G³ + 5G² - 2G - 2Compute each term:G ≈ 9.796G³ ≈ 9.796³ ≈ 9.796*9.796*9.796First compute 9.796² ≈ 95.96Then, 95.96*9.796 ≈ let's approximate:95.96*10 = 959.6Subtract 95.96*0.204 ≈ 19.56So, approximately 959.6 - 19.56 ≈ 940.04So, G³ ≈ 940.04Then, - (1/3)G³ ≈ -940.04 / 3 ≈ -313.3475G² ≈ 5*(95.96) ≈ 479.8-2G ≈ -2*9.796 ≈ -19.592-2 remains.So, total P(G) ≈ -313.347 + 479.8 -19.592 -2Compute step by step:-313.347 + 479.8 ≈ 166.453166.453 -19.592 ≈ 146.861146.861 -2 ≈ 144.861Which is approximately 144.861 million dollars, which matches our earlier calculation.So, the exact value is (214 + 46 sqrt(23))/3 million dollars, approximately 144.87 million dollars.Therefore, the amount of gold G that should be extracted to maximize profit is G = 5 + sqrt(23) tons, and the maximum profit is (214 + 46 sqrt(23))/3 million dollars.But let me write the exact form:G = 5 + sqrt(23)P(G) = (214 + 46 sqrt(23))/3Alternatively, factor numerator:214 + 46 sqrt(23) = 2*(107 + 23 sqrt(23))So, P(G) = 2*(107 + 23 sqrt(23))/3But both forms are acceptable.So, summarizing:1. The critical point of R(G) is at G = 0.3 tons, which is a local minimum.2. The amount of gold G that should be extracted to maximize profit is G = 5 + sqrt(23) tons, and the maximum profit is (214 + 46 sqrt(23))/3 million dollars.Final Answer1. The critical point is at ( G = boxed{0.3} ) tons, which is a local minimum.2. The amount of gold that should be extracted to maximize profit is ( G = boxed{5 + sqrt{23}} ) tons, and the maximum profit is ( boxed{dfrac{214 + 46sqrt{23}}{3}} ) million dollars.</think>