🧠:<think>Okay, so I have this problem about an ancient city with concentric rings, each representing different levels of political authority. The radius of each ring increases geometrically. Hmm, that sounds interesting. Let me try to figure this out step by step.First, part 1 asks me to derive a generalized formula for the area of the nth ring. The area of a ring is the area between two consecutive circles, right? So, if I have the radius of the nth ring, which is ( r_n = r_1 cdot k^{n-1} ), then the area of the nth ring would be the area of the circle with radius ( r_n ) minus the area of the circle with radius ( r_{n-1} ).Wait, actually, hold on. The nth ring is between the (n-1)th and nth circles. So, the area should be the area of the nth circle minus the area of the (n-1)th circle. Let me write that down.The area of the nth ring, ( A_n ), is:( A_n = pi r_n^2 - pi r_{n-1}^2 )Since ( r_n = r_1 cdot k^{n-1} ), then ( r_{n-1} = r_1 cdot k^{n-2} ). Plugging these into the area formula:( A_n = pi (r_1 k^{n-1})^2 - pi (r_1 k^{n-2})^2 )Simplify that:( A_n = pi r_1^2 k^{2(n-1)} - pi r_1^2 k^{2(n-2)} )Factor out ( pi r_1^2 k^{2(n-2)} ):( A_n = pi r_1^2 k^{2(n-2)} (k^2 - 1) )Hmm, that seems right. Let me check with n=1. If n=1, then ( A_1 ) should just be the area of the central palace, which is ( pi r_1^2 ). Plugging n=1 into the formula:( A_1 = pi r_1^2 k^{0} (k^2 - 1) = pi r_1^2 (k^2 - 1) )Wait, that doesn't match. Because when n=1, the area should just be ( pi r_1^2 ), but according to this, it's ( pi r_1^2 (k^2 - 1) ). That suggests my formula might be off.Let me think again. Maybe I made a mistake in the indices. The nth ring is between the (n-1)th and nth circles. So, for n=1, it's just the central palace, which is a circle, not a ring. So, actually, the first ring (n=1) is just the central palace, and the second ring (n=2) is the area between the first and second circles.Wait, maybe the problem defines the innermost ring as the central palace, so n=1 is the central palace, and n=2 is the first ring around it, and so on. So, in that case, the area of the nth ring would be:For n=1: ( A_1 = pi r_1^2 )For n>=2: ( A_n = pi (r_n^2 - r_{n-1}^2) )But in the problem statement, it says "the area of the nth ring is defined as the area enclosed between two consecutive circles." So, if n=1 is the central palace, then the first ring (n=1) is just the palace, and the second ring (n=2) is the area between n=1 and n=2 circles.Wait, that might be the confusion. Let me clarify. If the innermost ring is the central palace, which is a circle with radius r1, then the first ring (n=1) is just that circle. The second ring (n=2) is the area between r1 and r2. The third ring (n=3) is between r2 and r3, etc.So, in that case, the area of the nth ring for n>=1 is:If n=1: ( A_1 = pi r_1^2 )For n>=2: ( A_n = pi (r_n^2 - r_{n-1}^2) )But the problem says "the area of the nth ring is defined as the area enclosed between two consecutive circles." So, maybe n starts at 1, but the first ring is between r0 and r1? But r0 isn't defined. Hmm, perhaps the central palace is considered the 0th ring? But the problem says the innermost ring is r1.Wait, maybe the problem is considering the central palace as the first ring, so n=1 is the central palace, and n=2 is the first ring around it, etc. So, in that case, the area of the nth ring for n>=1 is:If n=1: ( A_1 = pi r_1^2 )For n>=2: ( A_n = pi (r_n^2 - r_{n-1}^2) )But in the problem statement, it's said that each subsequent ring's radius is increased by a factor of k, so the radius of the nth ring is ( r_n = r_1 cdot k^{n-1} ). So, for n=1, r1 is r1, n=2, r2=r1*k, n=3, r3=r1*k^2, etc.So, the area of the nth ring would be:For n=1: ( A_1 = pi r_1^2 )For n>=2: ( A_n = pi (r_1^2 k^{2(n-1)} - r_1^2 k^{2(n-2)}) )Which simplifies to:( A_n = pi r_1^2 (k^{2(n-1)} - k^{2(n-2)}) )Factor out ( k^{2(n-2)} ):( A_n = pi r_1^2 k^{2(n-2)} (k^2 - 1) )So, that seems consistent. But for n=1, plugging into this formula:( A_1 = pi r_1^2 k^{0} (k^2 - 1) = pi r_1^2 (k^2 - 1) )But that's not equal to ( pi r_1^2 ) unless k=1, which would make all rings have the same radius, which doesn't make sense. So, perhaps the formula is only valid for n>=2, and n=1 is a special case.So, the generalized formula for the area of the nth ring is:- For n=1: ( A_1 = pi r_1^2 )- For n>=2: ( A_n = pi r_1^2 k^{2(n-2)} (k^2 - 1) )Alternatively, we can write it as:( A_n = pi r_1^2 (k^{2(n-1)} - k^{2(n-2)}) )Which is the same as:( A_n = pi r_1^2 k^{2(n-2)} (k^2 - 1) )So, that's the formula for the area of the nth ring.Now, moving on to part 2. The archaeologist finds that the total area is 1,000,000 square units, the innermost radius r1 is 10 units, and there are 5 rings in total. We need to find k and verify if the total area matches.First, let's note that the total area is the sum of the areas of all 5 rings. So, total area ( A_{total} = A_1 + A_2 + A_3 + A_4 + A_5 )We have:- ( A_1 = pi r_1^2 = pi (10)^2 = 100pi )- ( A_2 = pi r_1^2 (k^2 - 1) = 100pi (k^2 - 1) )- ( A_3 = pi r_1^2 k^{2} (k^2 - 1) = 100pi k^{2} (k^2 - 1) )- ( A_4 = pi r_1^2 k^{4} (k^2 - 1) = 100pi k^{4} (k^2 - 1) )- ( A_5 = pi r_1^2 k^{6} (k^2 - 1) = 100pi k^{6} (k^2 - 1) )So, the total area is:( A_{total} = 100pi + 100pi (k^2 - 1) + 100pi k^{2} (k^2 - 1) + 100pi k^{4} (k^2 - 1) + 100pi k^{6} (k^2 - 1) )Let me factor out 100π:( A_{total} = 100pi [1 + (k^2 - 1) + k^2(k^2 - 1) + k^4(k^2 - 1) + k^6(k^2 - 1)] )Simplify inside the brackets:Let me denote ( S = 1 + (k^2 - 1) + k^2(k^2 - 1) + k^4(k^2 - 1) + k^6(k^2 - 1) )Let me compute S step by step:First term: 1Second term: ( (k^2 - 1) )Third term: ( k^2(k^2 - 1) = k^4 - k^2 )Fourth term: ( k^4(k^2 - 1) = k^6 - k^4 )Fifth term: ( k^6(k^2 - 1) = k^8 - k^6 )So, adding all these together:1 + (k^2 - 1) + (k^4 - k^2) + (k^6 - k^4) + (k^8 - k^6)Let me combine like terms:- The constant term: 1 -1 = 0- k^2 terms: k^2 - k^2 = 0- k^4 terms: k^4 - k^4 = 0- k^6 terms: k^6 - k^6 = 0- Remaining term: k^8So, S = k^8Wow, that's neat. So, the total area is:( A_{total} = 100pi cdot k^8 )But wait, the total area is given as 1,000,000 square units. So:( 100pi k^8 = 1,000,000 )Solve for k:Divide both sides by 100π:( k^8 = frac{1,000,000}{100pi} = frac{10,000}{pi} )So,( k = left( frac{10,000}{pi} right)^{1/8} )Let me compute that.First, compute 10,000 / π:10,000 / π ≈ 10,000 / 3.1416 ≈ 3183.0988618Now, take the 8th root of 3183.0988618.Let me compute that step by step.First, note that 3183.0988618 is approximately 3183.1.Let me compute the 8th root.We can write 3183.1 as approximately 3.1831 x 10^3.But maybe it's easier to compute using logarithms.Compute ln(3183.1) ≈ ln(3183.1) ≈ 8.066Then, ln(k) = (1/8) * ln(3183.1) ≈ (1/8)*8.066 ≈ 1.00825So, k ≈ e^{1.00825} ≈ 2.74Wait, let me check that.Alternatively, perhaps using exponents:We know that 2^8 = 2563^8 = 6561So, 3^8 is 6561, which is larger than 3183.1.So, k is between 2 and 3.Compute 2.7^8:2.7^2 = 7.292.7^4 = (7.29)^2 ≈ 53.14412.7^8 = (53.1441)^2 ≈ 2822.21Hmm, 2.7^8 ≈ 2822.21, which is less than 3183.1.Compute 2.8^8:2.8^2 = 7.842.8^4 = (7.84)^2 ≈ 61.46562.8^8 = (61.4656)^2 ≈ 3777.02So, 2.8^8 ≈ 3777.02, which is more than 3183.1.So, k is between 2.7 and 2.8.Let me try 2.75:2.75^2 = 7.56252.75^4 = (7.5625)^2 ≈ 57.19142.75^8 = (57.1914)^2 ≈ 3271.14That's closer. 3271.14 is still higher than 3183.1.Try 2.73:2.73^2 = 7.45292.73^4 = (7.4529)^2 ≈ 55.5452.73^8 = (55.545)^2 ≈ 3085.15Hmm, 3085.15 is less than 3183.1.So, between 2.73 and 2.75.Let me try 2.74:2.74^2 = 7.50762.74^4 = (7.5076)^2 ≈ 56.3642.74^8 = (56.364)^2 ≈ 3177.0That's very close to 3183.1.So, 2.74^8 ≈ 3177.0, which is just slightly less than 3183.1.So, k ≈ 2.74.Let me compute 2.74^8 more accurately.First, 2.74^2:2.74 * 2.74:2*2=4, 2*0.74=1.48, 0.74*2=1.48, 0.74*0.74=0.5476So, 4 + 1.48 + 1.48 + 0.5476 = 4 + 2.96 + 0.5476 = 7.5076So, 2.74^2 = 7.50762.74^4 = (7.5076)^2Compute 7.5076 * 7.5076:7*7=49, 7*0.5076=3.5532, 0.5076*7=3.5532, 0.5076*0.5076≈0.2576So, 49 + 3.5532 + 3.5532 + 0.2576 ≈ 49 + 7.1064 + 0.2576 ≈ 56.364So, 2.74^4 ≈ 56.3642.74^8 = (56.364)^2Compute 56.364 * 56.364:50*50=2500, 50*6.364=318.2, 6.364*50=318.2, 6.364*6.364≈40.5So, 2500 + 318.2 + 318.2 + 40.5 ≈ 2500 + 636.4 + 40.5 ≈ 3176.9So, 2.74^8 ≈ 3176.9, which is very close to 3183.1.The difference is 3183.1 - 3176.9 = 6.2So, to get a more accurate k, let's assume that the function k^8 is approximately linear near k=2.74.We have f(k) = k^8f(2.74) ≈ 3176.9We need f(k) = 3183.1So, delta_f = 3183.1 - 3176.9 = 6.2Compute f'(k) = 8k^7At k=2.74, f'(2.74) = 8*(2.74)^7First, compute 2.74^7:We know 2.74^2=7.50762.74^3=2.74*7.5076≈20.5892.74^4≈56.364 (as before)2.74^5=2.74*56.364≈154.332.74^6=2.74*154.33≈423.02.74^7=2.74*423.0≈1160.0So, f'(2.74)=8*1160≈9280So, delta_k ≈ delta_f / f'(k) ≈ 6.2 / 9280 ≈ 0.000668So, k ≈ 2.74 + 0.000668 ≈ 2.740668So, approximately 2.7407Therefore, k ≈ 2.7407Let me check with k=2.7407:Compute k^8:We can use the approximation:k^8 ≈ f(k) + f'(k)*delta_kBut since we already used that, maybe it's better to compute 2.7407^8.But that's time-consuming. Alternatively, since the difference is small, 2.7407 is approximately 2.74, and the error is minimal.So, k ≈ 2.74But let me check with k=2.7407:Compute 2.7407^2:2.7407*2.7407 ≈ (2.74)^2 + 2*2.74*0.0007 + (0.0007)^2 ≈ 7.5076 + 0.003836 + 0.00000049 ≈ 7.5114362.7407^4 = (7.511436)^2 ≈ 56.4212.7407^8 = (56.421)^2 ≈ 3183.0Yes, that's very close to 3183.1.So, k ≈ 2.7407Therefore, k ≈ 2.74So, the value of k is approximately 2.74.Now, let's verify if the total area matches.Given k ≈ 2.74, r1=10, total area should be 1,000,000.Compute total area:( A_{total} = 100pi k^8 )We have k^8 ≈ 3183.1, so:( A_{total} ≈ 100pi * 3183.1 ≈ 100 * 3.1416 * 3183.1 ≈ 3.1416 * 318310 ≈ )Wait, 100 * 3183.1 = 318,310So, 318,310 * π ≈ 318,310 * 3.1416 ≈Compute 318,310 * 3 = 954,930318,310 * 0.1416 ≈ 318,310 * 0.1 = 31,831318,310 * 0.04 = 12,732.4318,310 * 0.0016 ≈ 509.296So, total ≈ 31,831 + 12,732.4 + 509.296 ≈ 45,072.696So, total area ≈ 954,930 + 45,072.696 ≈ 1,000,002.696Which is approximately 1,000,002.7, very close to 1,000,000. The slight difference is due to rounding errors in k.Therefore, k ≈ 2.74 is correct, and the total area matches approximately.</think>

🧠:<think>Okay, so I have this problem about optimizing energy efficiency in smart homes for older people. It's divided into two parts. Let me try to tackle the first part first.The energy consumption is modeled by the function ( E(t) = A sin(omega t + phi) + C ). I need to find A, ω, φ, and C. They gave me that the total energy consumption over one day is 48 kWh. Also, the peak energy usage is at 6 PM with a maximum of 3 kW above the baseline. Hmm.First, let's recall that energy consumption over a day is the integral of the power over time. Since power is in kW and time is in hours, integrating over 24 hours should give me kWh. So, the total energy consumption is the integral from 0 to 24 of E(t) dt, which equals 48 kWh.Let me write that down:[int_{0}^{24} E(t) , dt = 48]Substituting E(t):[int_{0}^{24} left( A sin(omega t + phi) + C right) dt = 48]I can split this integral into two parts:[int_{0}^{24} A sin(omega t + phi) , dt + int_{0}^{24} C , dt = 48]The integral of a sine function over a full period is zero. Since it's a 24-hour period, and the function is periodic, I think ω is such that the period is 24 hours. So, ω = 2π / 24 = π / 12 rad/hour. Let me confirm that.The period T is 24 hours, so ω = 2π / T = 2π / 24 = π / 12. Yes, that seems right.So, the integral of the sine part over 0 to 24 is zero. Therefore, the first integral is zero, and the second integral is C * 24. So:[C * 24 = 48 implies C = 48 / 24 = 2 text{ kW}]Okay, so C is 2 kW. That's the baseline energy consumption.Next, they mentioned that the peak energy usage is at 6 PM with a maximum of 3 kW above the baseline. So, the maximum E(t) is C + A = 2 + 3 = 5 kW. Therefore, A must be 3 kW.So, A = 3 kW.Now, we need to find ω and φ. We already have ω as π / 12 rad/hour. So, that's determined.Now, the phase shift φ. The peak occurs at 6 PM. Let's assume t is measured in hours, starting from midnight. So, 6 PM is t = 18 hours.At t = 18, E(t) is maximum. The sine function reaches its maximum when its argument is π/2. So,[omega t + phi = pi/2]At t = 18:[(pi / 12) * 18 + phi = pi / 2]Let me compute (π / 12) * 18:18 / 12 = 1.5, so 1.5π.So,1.5π + φ = π / 2Solving for φ:φ = π / 2 - 1.5π = π / 2 - 3π / 2 = (-2π)/2 = -πSo, φ = -π radians.Let me check if that makes sense. So, the sine function is shifted by -π, which is equivalent to shifting it by π to the right, because phase shift is -φ / ω. Wait, actually, in the function sin(ω t + φ), the phase shift is -φ / ω. So, if φ is -π, then the phase shift is π / (π / 12) = 12 hours. So, the sine wave is shifted 12 hours to the right.But wait, the maximum occurs at t = 18. Let's see, without any phase shift, the sine function sin(ω t) would have its maximum at t = (π/2) / ω. Since ω = π / 12, that would be at t = (π / 2) / (π / 12) = (π / 2) * (12 / π) = 6 hours. So, at t = 6, which is 6 AM. But we need the maximum at t = 18, which is 12 hours later. So, shifting the sine wave by 12 hours to the right would make the maximum occur at t = 18. So, that makes sense.Therefore, φ = -π.So, summarizing:A = 3 kWω = π / 12 rad/hourφ = -π radiansC = 2 kWLet me double-check.E(t) = 3 sin( (π / 12) t - π ) + 2At t = 18:E(18) = 3 sin( (π / 12)*18 - π ) + 2 = 3 sin( 1.5π - π ) + 2 = 3 sin(0.5π) + 2 = 3*1 + 2 = 5 kW. That's correct.Also, the total energy over 24 hours is 24*C = 24*2 = 48 kWh, which matches. So, that seems good.Now, moving on to part 2.We need to install a solar panel system that generates power according to P(t) = B sin(ω t + θ). We need to find B and θ such that the difference between E(t) and P(t) is minimized throughout the day. The peak solar power is 2 kW at 12 PM.So, similar to part 1, the solar power function is also a sine wave with the same angular frequency ω = π / 12, but different amplitude B and phase shift θ.The goal is to minimize the difference between E(t) and P(t). I think this could be interpreted in a few ways. Maybe minimize the integral of the absolute difference, or minimize the maximum difference, or minimize the squared difference. Since it's about optimization, perhaps we need to minimize the integral of the squared difference over the day, which would be a least squares approach.But let's see what information is given. The peak solar power is 2 kW at 12 PM. So, similar to part 1, the maximum of P(t) is B, and it occurs at t = 12.So, similar to part 1, the maximum of P(t) occurs when the argument of sine is π/2. So,ω t + θ = π / 2 at t = 12.So,(π / 12)*12 + θ = π / 2Simplify:π + θ = π / 2Therefore, θ = π / 2 - π = -π / 2.So, θ = -π / 2.Therefore, P(t) = B sin( (π / 12) t - π / 2 )But we also know that the peak solar power is 2 kW, so the amplitude B is 2 kW.Wait, but let me think. The maximum of P(t) is B, so B = 2 kW.So, P(t) = 2 sin( (π / 12) t - π / 2 )Alternatively, we can write this as -2 cos( (π / 12) t ), since sin(x - π/2) = -cos(x). So, P(t) = -2 cos( (π / 12) t )But let's see if that makes sense. At t = 12, P(t) should be 2 kW.Compute P(12):P(12) = 2 sin( (π / 12)*12 - π / 2 ) = 2 sin( π - π / 2 ) = 2 sin( π / 2 ) = 2*1 = 2 kW. Correct.So, that's good.But now, the goal is to minimize the difference between E(t) and P(t). So, we need to define what "minimize the difference" means. If it's just matching the peak, then we've already done that. But probably, it's about minimizing some measure of the difference over the entire day.Assuming we need to minimize the integral of (E(t) - P(t))^2 dt over 0 to 24, which is a common approach in optimization.So, let's set up the integral:[int_{0}^{24} (E(t) - P(t))^2 dt]We need to find B and θ such that this integral is minimized. But wait, in the problem statement, it says "the solar panel system that generates power according to the function P(t) = B sin(ω t + θ)", and we need to determine B and θ. However, they also say the peak solar power is 2 kW at 12 PM, which gives us B = 2 and θ = -π / 2 as above.But wait, if we already have B and θ determined by the peak, then maybe the optimization is already done? Or perhaps, the problem is that the solar power is supposed to match the energy consumption as closely as possible, so maybe we need to adjust B and θ not just to match the peak, but to align the two functions in a way that their difference is minimized.Wait, but the problem says "the goal is to minimize the difference between energy consumption and solar power generation throughout the day. Assuming the peak solar power generation is 2 kW at 12 PM, calculate the optimal values of B and θ to achieve this goal."So, they are giving us that the peak solar power is 2 kW at 12 PM, so that gives us B = 2 and θ = -π / 2 as above. But perhaps we need to adjust θ further to align the solar power with the energy consumption.Wait, but in part 1, E(t) has a phase shift of -π, meaning it's shifted 12 hours to the right, so its maximum is at 18 (6 PM). The solar power is shifted to have maximum at 12 (noon). So, the two functions are 6 hours apart in their peaks.So, perhaps to minimize the difference, we need to shift the solar power function so that it aligns better with the energy consumption function.Wait, but the problem says that the peak solar power is at 12 PM, so θ is fixed as -π / 2. So, we can't change θ anymore because the peak is fixed at 12 PM. So, maybe we can only adjust B? But B is given by the peak power, which is 2 kW. So, perhaps B is fixed as 2, and θ is fixed as -π / 2. So, we can't adjust them anymore.But the problem says "calculate the optimal values of B and θ to achieve this goal." So, maybe we need to adjust both B and θ such that the peak is 2 kW at 12 PM, and also the functions are aligned to minimize the difference.Wait, but if we adjust θ, the peak time changes. So, if we fix the peak at 12 PM, θ is fixed. So, perhaps B is also fixed as 2. So, maybe the answer is B = 2 and θ = -π / 2.But let me think again. Maybe the problem is that the solar power is supposed to match the energy consumption as much as possible, so perhaps we need to adjust θ such that the solar power is in phase with the energy consumption, but with the constraint that the peak is at 12 PM.Wait, but if the energy consumption peaks at 6 PM, and solar power peaks at 12 PM, they are 6 hours apart. So, maybe we can't align them perfectly, but perhaps we can adjust θ to make the solar power function as close as possible to the energy consumption function.Wait, but the problem says "the goal is to minimize the difference between energy consumption and solar power generation throughout the day." So, perhaps we need to find B and θ such that the integral of |E(t) - P(t)| is minimized, subject to P(t) having a peak of 2 kW at 12 PM.Alternatively, maybe we can model this as a least squares problem where we minimize the integral of (E(t) - P(t))^2 dt, with the constraints that P(t) has a peak of 2 kW at 12 PM.But in that case, we can set up the problem with B and θ as variables, but with the constraint that P(t) peaks at 12 PM with 2 kW.Wait, but the peak condition gives us two equations: the maximum of P(t) is 2, and it occurs at t = 12. So, that gives us B = 2 and θ = -π / 2 as before.So, perhaps the problem is just to set B = 2 and θ = -π / 2, because the peak is fixed at 12 PM with 2 kW.But maybe I'm missing something. Let me think again.Alternatively, perhaps the solar power function can be adjusted not just to have a peak at 12 PM, but also to align with the energy consumption function in some way. For example, maybe shifting θ so that the solar power is in phase with the energy consumption, but still peaking at 12 PM.Wait, but if the energy consumption is shifted by -π, which is 12 hours, and the solar power is shifted by -π / 2, which is 6 hours, then the solar power is 6 hours ahead of the energy consumption.So, maybe to minimize the difference, we need to shift the solar power function so that it aligns better with the energy consumption.But the problem states that the peak solar power is at 12 PM, so we can't shift it beyond that. So, perhaps the optimal θ is -π / 2, as we found, and B is 2.Alternatively, maybe we can adjust θ to make the solar power function as close as possible to the energy consumption function, but keeping the peak at 12 PM.Wait, perhaps we can model this as a phase shift problem.Let me write E(t) and P(t):E(t) = 3 sin( (π / 12) t - π ) + 2P(t) = 2 sin( (π / 12) t + θ )We need to find θ such that the integral of (E(t) - P(t))^2 is minimized.But since E(t) is a sine wave plus a constant, and P(t) is a sine wave, perhaps we can express E(t) as a sine wave plus a constant, and then find the best fit for P(t) to match E(t).But E(t) is 3 sin( (π / 12) t - π ) + 2, which is equivalent to -3 sin( (π / 12) t ) + 2.So, E(t) = -3 sin( (π / 12) t ) + 2And P(t) = 2 sin( (π / 12) t + θ )We need to find θ to minimize the integral of (E(t) - P(t))^2 dt.Let me set up the integral:[int_{0}^{24} left( -3 sinleft( frac{pi}{12} t right) + 2 - 2 sinleft( frac{pi}{12} t + theta right) right)^2 dt]This seems complicated, but maybe we can use some trigonometric identities to simplify it.First, let me denote ω = π / 12 for simplicity.So, E(t) = -3 sin(ω t) + 2P(t) = 2 sin(ω t + θ )So, the difference is:E(t) - P(t) = -3 sin(ω t) + 2 - 2 sin(ω t + θ )Let me expand sin(ω t + θ ) using the sine addition formula:sin(ω t + θ ) = sin(ω t) cosθ + cos(ω t) sinθSo, P(t) = 2 [ sin(ω t) cosθ + cos(ω t) sinθ ]Therefore, E(t) - P(t) becomes:-3 sin(ω t) + 2 - 2 sin(ω t) cosθ - 2 cos(ω t) sinθCombine like terms:[ -3 - 2 cosθ ] sin(ω t) - 2 sinθ cos(ω t) + 2So, the difference is:[ -3 - 2 cosθ ] sin(ω t) - 2 sinθ cos(ω t) + 2Now, let's square this difference:[ -3 - 2 cosθ ]^2 sin^2(ω t) + [ -2 sinθ ]^2 cos^2(ω t) + 2 * [ -3 - 2 cosθ ] * [ -2 sinθ ] sin(ω t) cos(ω t) + 2^2 + 2 * [ -3 - 2 cosθ ] * 2 sin(ω t) + 2 * [ -2 sinθ ] * 2 cos(ω t)Wait, this is getting too complicated. Maybe instead of expanding everything, I can use orthogonality of sine and cosine functions over the interval.Since we're integrating over a full period (24 hours), the integrals of sin(ω t) and cos(ω t) over 0 to 24 are zero. Also, the integrals of sin^2 and cos^2 over a full period are equal to half the period.So, let's compute the integral:[int_{0}^{24} left( a sin(omega t) + b cos(omega t) + c right)^2 dt]Expanding this:[int_{0}^{24} (a^2 sin^2(omega t) + b^2 cos^2(omega t) + c^2 + 2ab sin(omega t)cos(omega t) + 2ac sin(omega t) + 2bc cos(omega t)) dt]Now, integrating term by term:1. ( int_{0}^{24} a^2 sin^2(omega t) dt = a^2 * (24 / 2) = 12 a^2 )2. ( int_{0}^{24} b^2 cos^2(omega t) dt = b^2 * (24 / 2) = 12 b^2 )3. ( int_{0}^{24} c^2 dt = 24 c^2 )4. ( int_{0}^{24} 2ab sin(omega t)cos(omega t) dt = ab * int_{0}^{24} sin(2ω t) dt = 0 ) because it's a full period.5. ( int_{0}^{24} 2ac sin(omega t) dt = 0 )6. ( int_{0}^{24} 2bc cos(omega t) dt = 0 )So, the integral simplifies to:12 a^2 + 12 b^2 + 24 c^2In our case, the difference E(t) - P(t) is:[ -3 - 2 cosθ ] sin(ω t) - 2 sinθ cos(ω t) + 2So, comparing to the general form a sin(ω t) + b cos(ω t) + c, we have:a = -3 - 2 cosθb = -2 sinθc = 2Therefore, the integral becomes:12 a^2 + 12 b^2 + 24 c^2Substituting a, b, c:12 (-3 - 2 cosθ)^2 + 12 (-2 sinθ)^2 + 24 (2)^2Let me compute each term:First term: 12 (-3 - 2 cosθ)^2Let me expand (-3 - 2 cosθ)^2:= 9 + 12 cosθ + 4 cos²θSo, first term: 12*(9 + 12 cosθ + 4 cos²θ) = 108 + 144 cosθ + 48 cos²θSecond term: 12*(-2 sinθ)^2 = 12*4 sin²θ = 48 sin²θThird term: 24*(4) = 96So, total integral:108 + 144 cosθ + 48 cos²θ + 48 sin²θ + 96Combine like terms:108 + 96 = 204144 cosθ remains48 cos²θ + 48 sin²θ = 48 (cos²θ + sin²θ) = 48*1 = 48So, total integral:204 + 144 cosθ + 48 = 252 + 144 cosθTherefore, the integral of (E(t) - P(t))^2 dt = 252 + 144 cosθWe need to minimize this integral with respect to θ.So, the integral is 252 + 144 cosθ. To minimize this, we need to minimize cosθ, which occurs when cosθ is as small as possible, i.e., cosθ = -1.Therefore, the minimum integral is 252 + 144*(-1) = 252 - 144 = 108.So, the minimum occurs when cosθ = -1, which implies θ = π radians.But wait, earlier we had that θ = -π / 2 to have the peak at 12 PM. So, is there a conflict here?Wait, no. Because in this part, we are trying to find θ to minimize the integral, but we have a constraint that the peak solar power is 2 kW at 12 PM, which requires θ = -π / 2.So, perhaps we cannot choose θ freely. We have to set θ = -π / 2 to satisfy the peak condition, and then compute B accordingly.Wait, but in the problem statement, it says "the goal is to minimize the difference between energy consumption and solar power generation throughout the day. Assuming the peak solar power generation is 2 kW at 12 PM, calculate the optimal values of B and θ to achieve this goal."So, perhaps B is not fixed at 2, but we can adjust B and θ such that the peak is 2 kW at 12 PM, and also the integral is minimized.Wait, but if we fix the peak at 12 PM as 2 kW, then B must be 2, and θ must be -π / 2, as we found earlier.But then, the integral of the squared difference would be 252 + 144 cosθ, with θ = -π / 2.Compute cos(-π / 2) = 0.So, the integral becomes 252 + 144*0 = 252.But earlier, when we tried to minimize it by setting θ = π, we got a lower integral of 108, but that would shift the peak to a different time, which contradicts the given condition.So, perhaps the problem is that we have to fix θ = -π / 2 to have the peak at 12 PM, and then B is fixed at 2. Therefore, the optimal values are B = 2 and θ = -π / 2.But wait, maybe I'm misunderstanding the problem. Maybe the solar power function can have a different amplitude and phase, but still have a peak of 2 kW at 12 PM. So, perhaps B is not necessarily 2, but the peak is 2, which occurs when the sine function is at its maximum, so B must be 2. So, B is fixed at 2, and θ is fixed at -π / 2.Therefore, the optimal values are B = 2 and θ = -π / 2.Alternatively, maybe the problem allows for B to be different, but the peak is 2, so B is 2, and θ is -π / 2.Wait, but in that case, the integral is 252, which is higher than the minimum possible 108, but we have to satisfy the peak condition.So, perhaps the answer is B = 2 and θ = -π / 2.Alternatively, maybe we can adjust B and θ such that the solar power function is as close as possible to the energy consumption function, while still having a peak of 2 kW at 12 PM.Wait, but if we adjust B and θ, the peak condition gives us two equations:1. The maximum of P(t) is 2, so B = 2.2. The maximum occurs at t = 12, so θ = -π / 2.Therefore, B and θ are uniquely determined by the peak condition, so we can't adjust them further.Therefore, the optimal values are B = 2 and θ = -π / 2.But let me check if that makes sense. If we set θ = -π / 2, then P(t) = 2 sin( (π / 12) t - π / 2 ) = -2 cos( (π / 12) t )And E(t) = -3 sin( (π / 12) t ) + 2So, the difference E(t) - P(t) = -3 sin(ω t) + 2 - (-2 cos(ω t)) = -3 sin(ω t) + 2 + 2 cos(ω t)So, the difference is -3 sin(ω t) + 2 cos(ω t) + 2The integral of the square of this is 252, as we found earlier.Alternatively, if we set θ = π, then P(t) = 2 sin( (π / 12) t + π ) = -2 sin( (π / 12) t )Then, the difference E(t) - P(t) = -3 sin(ω t) + 2 - (-2 sin(ω t)) = (-3 + 2) sin(ω t) + 2 = -sin(ω t) + 2The integral of the square would be:[int_{0}^{24} (-sin(ω t) + 2)^2 dt = int_{0}^{24} sin^2(ω t) - 4 sin(ω t) + 4 dt]Which is:12*(1/2) + 0 + 24*4 = 6 + 96 = 102Wait, that's even less than 108. But wait, this would mean that the solar power function is shifted by π, which would make the peak at t = (π / 2 - θ)/ω = (π / 2 - π)/ (π / 12) = (-π / 2) / (π / 12) = -6 hours, which is not within our 24-hour period. So, the peak would be at t = 18 (6 PM), but we need it at t = 12 (noon). So, we can't set θ = π because that would shift the peak to 6 PM, which is not desired.Therefore, we have to stick with θ = -π / 2 to have the peak at 12 PM, which gives us B = 2 and θ = -π / 2.So, in conclusion, the optimal values are B = 2 kW and θ = -π / 2 radians.But let me double-check the integral with θ = -π / 2.E(t) - P(t) = -3 sin(ω t) + 2 - (-2 cos(ω t)) = -3 sin(ω t) + 2 + 2 cos(ω t)So, the integral is:[int_{0}^{24} (-3 sin(ω t) + 2 + 2 cos(ω t))^2 dt]Expanding this:= int_{0}^{24} [9 sin²(ω t) + 4 + 4 cos²(ω t) + 12 sin(ω t) + 8 cos(ω t) - 12 sin(ω t) cos(ω t)] dtBut wait, actually, expanding (-3 sin + 2 + 2 cos)^2:= 9 sin² + 4 + 4 cos² + 12 sin + 8 cos - 12 sin cosWait, no, let's do it properly.(-3 sin + 2 + 2 cos)^2 = (-3 sin)^2 + (2)^2 + (2 cos)^2 + 2*(-3 sin)*(2) + 2*(-3 sin)*(2 cos) + 2*(2)*(2 cos)= 9 sin² + 4 + 4 cos² - 12 sin - 12 sin cos + 8 cosSo, the integral becomes:9 ∫ sin² dt + 4 ∫ dt + 4 ∫ cos² dt - 12 ∫ sin dt - 12 ∫ sin cos dt + 8 ∫ cos dtNow, over 0 to 24:∫ sin² dt = 12∫ cos² dt = 12∫ sin dt = 0∫ cos dt = 0∫ sin cos dt = 0So, the integral is:9*12 + 4*24 + 4*12 - 12*0 - 12*0 + 8*0 = 108 + 96 + 48 = 252Which matches our earlier result.So, yes, the integral is 252 with θ = -π / 2.If we tried to adjust θ to minimize the integral without the peak constraint, we could get a lower value, but since the peak is fixed at 12 PM, we have to accept this higher integral.Therefore, the optimal values are B = 2 kW and θ = -π / 2 radians.</think>

🧠:<think>Okay, so I have this problem about a coding academy in Techville, and I need to figure out the long-term impact on the workforce and investment levels. The problem is split into two parts, each with its own differential equation. Let me try to tackle them one by one.Starting with the first part: the rate of change of the number of skilled tech workers, G(t), is given by the differential equation:[frac{dG}{dt} = k_1 G(t) left(1 - frac{G(t)}{M}right) - c I(t)]They mention that I(t) is a constant, I₀. So, substituting that in, the equation becomes:[frac{dG}{dt} = k_1 G(t) left(1 - frac{G(t)}{M}right) - c I_0]This looks like a logistic growth model with a constant term subtracted. The logistic part is the first term, which models growth with a carrying capacity M, and then there's a constant outflow term, -c I₀, which might represent some kind of loss or maybe the effect of investment on the number of graduates. Hmm, actually, in the problem statement, it says c is the rate at which investment inflow affects the number of graduates. So, maybe it's not an outflow but a term that subtracts from the growth rate.Anyway, the equation is a nonlinear differential equation because of the G(t) squared term. To solve this, I think I can rewrite it as:[frac{dG}{dt} = k_1 G(t) - frac{k_1}{M} G(t)^2 - c I_0]This is a Riccati equation, which is a type of nonlinear differential equation. Riccati equations can sometimes be transformed into linear equations through substitution. Let me see if I can do that here.Let me rearrange the equation:[frac{dG}{dt} + frac{k_1}{M} G(t)^2 - k_1 G(t) + c I_0 = 0]Hmm, actually, maybe I can write it in standard Riccati form:[frac{dG}{dt} = a G(t)^2 + b G(t) + c]Comparing, we have:a = -k₁/Mb = k₁c = -c I₀So, yes, it's a Riccati equation. The standard Riccati equation is:[frac{dy}{dt} = q_0(t) + q_1(t) y + q_2(t) y^2]In our case, q₀(t) = -c I₀, q₁(t) = k₁, q₂(t) = -k₁/M.I remember that Riccati equations can sometimes be linearized if we know a particular solution. But I don't have a particular solution here. Maybe I can use substitution to make it linear.Let me try the substitution:Let y = 1/G(t). Then, dy/dt = - (1/G²) dG/dt.Substituting into the equation:- (1/G²) dG/dt = -c I₀ + k₁ G - (k₁/M) G²Multiply both sides by -G²:dG/dt = c I₀ G² - k₁ G³ + (k₁/M) G⁴Wait, that seems more complicated. Maybe that substitution isn't helpful.Alternatively, maybe I can use an integrating factor or another substitution. Let me think.Alternatively, since it's a Riccati equation, maybe I can use the substitution:Let G(t) = u(t) + v(t), where u(t) is a particular solution and v(t) is the homogeneous solution. But I don't know a particular solution yet.Alternatively, maybe I can use the substitution z = 1/(G(t) - A), where A is a constant to be determined. Let me try that.Let z = 1/(G - A). Then, dz/dt = - (1/(G - A)²) dG/dt.Substituting into the differential equation:- (1/(G - A)²) dG/dt = -c I₀ + k₁ G - (k₁/M) G²But dG/dt is equal to k₁ G (1 - G/M) - c I₀, so:- (1/(G - A)²) [k₁ G (1 - G/M) - c I₀] = -c I₀ + k₁ G - (k₁/M) G²This seems messy. Maybe another approach.Alternatively, let me consider that this is a Bernoulli equation. Wait, no, Bernoulli equations have the form dy/dt + P(t) y = Q(t) y^n. Our equation is:dG/dt + (k₁/M) G² - k₁ G + c I₀ = 0Which is similar but not exactly the standard Bernoulli form. Maybe rearrange terms:dG/dt = - (k₁/M) G² + k₁ G - c I₀So, it's:dG/dt + (k₁/M) G² - k₁ G + c I₀ = 0Hmm, maybe I can write it as:dG/dt = - (k₁/M) G² + k₁ G - c I₀This is a quadratic in G. Maybe I can complete the square or find an integrating factor.Alternatively, maybe I can use separation of variables. Let me try that.Rewrite the equation as:dG / [ - (k₁/M) G² + k₁ G - c I₀ ] = dtSo, integrating both sides:∫ [ - (k₁/M) G² + k₁ G - c I₀ ]^{-1} dG = ∫ dtThis integral might be complicated, but perhaps we can factor the denominator or use partial fractions.Let me write the denominator as:- (k₁/M) G² + k₁ G - c I₀ = - (k₁/M) [ G² - M G + (M c I₀)/k₁ ]So, factor out -k₁/M:= - (k₁/M) [ G² - M G + (M c I₀)/k₁ ]Let me denote the quadratic inside as Q(G) = G² - M G + (M c I₀)/k₁We can compute the discriminant of Q(G):Δ = M² - 4 * 1 * (M c I₀)/k₁ = M² - (4 M c I₀)/k₁Depending on the value of Δ, we can factor Q(G) or write it in terms of its roots.Case 1: Δ > 0, two real roots.Case 2: Δ = 0, repeated real root.Case 3: Δ < 0, complex roots.Let me assume Δ ≠ 0 for now, so we can factor Q(G).Let me denote the roots as G₁ and G₂:G₁, G₂ = [ M ± sqrt(Δ) ] / 2So, Q(G) = (G - G₁)(G - G₂)Therefore, the denominator becomes:- (k₁/M) (G - G₁)(G - G₂)So, the integral becomes:∫ [ - (k₁/M) (G - G₁)(G - G₂) ]^{-1} dG = ∫ dtWhich simplifies to:∫ [ - M/(k₁) * 1/( (G - G₁)(G - G₂) ) ]^{-1} dG = ∫ dtWait, no, let me correct that.The integral is:∫ [ - (k₁/M) (G - G₁)(G - G₂) ]^{-1} dG = ∫ dtWhich is:∫ [ - M/(k₁) * 1/( (G - G₁)(G - G₂) ) ] dG = ∫ dtSo, factor out constants:- M/k₁ ∫ [ 1/( (G - G₁)(G - G₂) ) ] dG = ∫ dtNow, we can perform partial fractions on 1/( (G - G₁)(G - G₂) )Let me write:1/( (G - G₁)(G - G₂) ) = A/(G - G₁) + B/(G - G₂)Solving for A and B:1 = A (G - G₂) + B (G - G₁)Set G = G₁:1 = A (G₁ - G₂) => A = 1/(G₁ - G₂)Similarly, set G = G₂:1 = B (G₂ - G₁) => B = 1/(G₂ - G₁) = -1/(G₁ - G₂)So, A = 1/(G₁ - G₂), B = -1/(G₁ - G₂)Therefore, the integral becomes:- M/k₁ ∫ [ A/(G - G₁) + B/(G - G₂) ] dG = ∫ dtSubstituting A and B:= - M/k₁ ∫ [ 1/(G₁ - G₂) * (1/(G - G₁) - 1/(G - G₂)) ] dG= - M/(k₁ (G₁ - G₂)) ∫ [ 1/(G - G₁) - 1/(G - G₂) ] dGIntegrate term by term:= - M/(k₁ (G₁ - G₂)) [ ln|G - G₁| - ln|G - G₂| ] + C = tSimplify the logarithms:= - M/(k₁ (G₁ - G₂)) ln| (G - G₁)/(G - G₂) | + C = tLet me write this as:ln| (G - G₁)/(G - G₂) | = - (k₁ (G₁ - G₂)/M) (t - C)Exponentiate both sides:| (G - G₁)/(G - G₂) | = e^{ - (k₁ (G₁ - G₂)/M) (t - C) }Let me denote K = e^{- (k₁ (G₁ - G₂)/M) C }, which is just another constant.So,(G - G₁)/(G - G₂) = ± K e^{ - (k₁ (G₁ - G₂)/M) t }Let me combine constants into a single constant C:(G - G₁)/(G - G₂) = C e^{ - (k₁ (G₁ - G₂)/M) t }Now, solving for G:Let me denote the exponent coefficient as:λ = - (k₁ (G₁ - G₂)/M )So,(G - G₁)/(G - G₂) = C e^{ λ t }Cross-multiplying:G - G₁ = C e^{ λ t } (G - G₂ )Bring all G terms to one side:G - G₁ = C e^{ λ t } G - C e^{ λ t } G₂G - C e^{ λ t } G = G₁ - C e^{ λ t } G₂Factor G:G (1 - C e^{ λ t }) = G₁ - C e^{ λ t } G₂Therefore,G(t) = [ G₁ - C e^{ λ t } G₂ ] / [ 1 - C e^{ λ t } ]Now, we can apply the initial condition G(0) = G₀.At t=0:G(0) = [ G₁ - C G₂ ] / [ 1 - C ] = G₀Solve for C:G₀ (1 - C ) = G₁ - C G₂G₀ - G₀ C = G₁ - C G₂Bring terms with C to one side:- G₀ C + C G₂ = G₁ - G₀Factor C:C ( - G₀ + G₂ ) = G₁ - G₀Thus,C = (G₁ - G₀)/(G₂ - G₀ )So, substituting back into G(t):G(t) = [ G₁ - ( (G₁ - G₀)/(G₂ - G₀ ) ) e^{ λ t } G₂ ] / [ 1 - ( (G₁ - G₀)/(G₂ - G₀ ) ) e^{ λ t } ]This expression can be simplified, but it's already quite involved. Let me see if I can express it differently.Alternatively, since G₁ and G₂ are roots of the quadratic equation Q(G) = 0, which is G² - M G + (M c I₀)/k₁ = 0, we can express G₁ and G₂ in terms of M, c, I₀, and k₁.Recall that:G₁ + G₂ = MG₁ G₂ = (M c I₀)/k₁So, perhaps we can express the solution in terms of these.But maybe it's better to leave it in terms of G₁ and G₂ for now.So, summarizing, the solution for G(t) is:G(t) = [ G₁ - C e^{ λ t } G₂ ] / [ 1 - C e^{ λ t } ]where C = (G₁ - G₀)/(G₂ - G₀ )and λ = - (k₁ (G₁ - G₂)/M )Alternatively, since G₁ and G₂ are roots, we can write the solution in terms of the initial condition and the roots.But this seems quite involved. Maybe I can consider specific cases or see if the solution can be expressed in a more compact form.Alternatively, perhaps I can write the solution as:G(t) = [ (G₁ - G₂) e^{ λ t } + G₂ ] / [ (G₁ - G₂) e^{ λ t } + G₁ - G₂ ]Wait, let me check that.Wait, from the expression:G(t) = [ G₁ - C e^{ λ t } G₂ ] / [ 1 - C e^{ λ t } ]Let me factor out G₁ in the numerator and denominator:= [ G₁ (1 - (C G₂ / G₁) e^{ λ t }) ] / [ 1 - C e^{ λ t } ]But unless C G₂ / G₁ is something specific, this might not help.Alternatively, let me write it as:G(t) = [ G₁ - C G₂ e^{ λ t } ] / [ 1 - C e^{ λ t } ]Let me denote D = C G₂, then:G(t) = [ G₁ - D e^{ λ t } ] / [ 1 - (D / G₂) e^{ λ t } ]But I don't know if that helps.Alternatively, let me consider the behavior as t approaches infinity. Since the mayor is interested in the long-term behavior, maybe I can find the equilibrium points and see if G(t) approaches them.Looking back at the differential equation:dG/dt = k₁ G (1 - G/M ) - c I₀Set dG/dt = 0:k₁ G (1 - G/M ) - c I₀ = 0This is a quadratic equation in G:k₁ G - (k₁/M) G² - c I₀ = 0Multiply through by -M:(k₁/M) G² - k₁ G + c I₀ M = 0Wait, no, let me rearrange:(k₁/M) G² - k₁ G + c I₀ = 0Multiply both sides by M:k₁ G² - k₁ M G + c I₀ M = 0So,k₁ G² - k₁ M G + c I₀ M = 0This quadratic equation will have solutions:G = [ k₁ M ± sqrt( (k₁ M)^2 - 4 k₁ (c I₀ M) ) ] / (2 k₁ )Simplify discriminant:Δ = (k₁ M)^2 - 4 k₁ (c I₀ M ) = k₁² M² - 4 k₁ c I₀ MFactor out k₁ M:Δ = k₁ M (k₁ M - 4 c I₀ )So, for real solutions, we need Δ ≥ 0:k₁ M (k₁ M - 4 c I₀ ) ≥ 0Assuming k₁ and M are positive (as they are growth rates and maximum positions), this implies:k₁ M - 4 c I₀ ≥ 0 => I₀ ≤ (k₁ M)/ (4 c )So, if I₀ is less than or equal to (k₁ M)/(4 c ), we have two equilibrium points. Otherwise, no real equilibria, meaning G(t) will approach infinity or negative infinity, but since G(t) represents the number of skilled workers, it can't be negative, so it would approach infinity.But in reality, G(t) can't exceed M because of the logistic term, but the subtraction of c I₀ complicates things.Wait, actually, the logistic term is k₁ G (1 - G/M ), which is positive when G < M and negative when G > M, tending to limit G to M. However, the subtraction of c I₀ could potentially cause G(t) to decrease even when G < M.So, the equilibria are at G where k₁ G (1 - G/M ) = c I₀So, if c I₀ is too large, the quadratic equation may not have real roots, meaning G(t) will not stabilize and may decrease indefinitely, but since G(t) can't be negative, perhaps it will approach zero or some other behavior.But in our case, since I(t) is constant, I₀, we can have different scenarios.However, the problem says to determine the expression for G(t) given I(t) is constant I₀. So, regardless of the equilibria, we need the general solution.Given that, and considering the complexity of the solution, perhaps it's better to express it in terms of the roots G₁ and G₂, as we did earlier.So, summarizing, the solution is:G(t) = [ G₁ - C e^{ λ t } G₂ ] / [ 1 - C e^{ λ t } ]where C = (G₁ - G₀)/(G₂ - G₀ )and λ = - (k₁ (G₁ - G₂)/M )Alternatively, since G₁ and G₂ are roots of the quadratic equation, we can express them as:G₁, G₂ = [ M ± sqrt(M² - 4 (c I₀ M)/k₁ ) ] / 2So, G₁ = [ M + sqrt(M² - 4 c I₀ M / k₁ ) ] / 2G₂ = [ M - sqrt(M² - 4 c I₀ M / k₁ ) ] / 2Therefore, the solution can be written in terms of M, c, I₀, k₁, and the initial condition G₀.But this is quite involved, and I'm not sure if there's a simpler way to express it. Maybe we can consider the case where I₀ is small enough that the discriminant is positive, so we have two real roots, and G(t) approaches one of them as t increases.Alternatively, perhaps we can write the solution in terms of hyperbolic functions or something similar, but I think the form we have is acceptable.So, for the first part, the expression for G(t) is:G(t) = [ G₁ - C e^{ λ t } G₂ ] / [ 1 - C e^{ λ t } ]where G₁ and G₂ are the roots of the quadratic equation k₁ G² - k₁ M G + c I₀ M = 0, and C is determined by the initial condition G(0) = G₀.Alternatively, we can write it as:G(t) = (G₁ - G₂) / [ 1 + ( (G₁ - G₀)/(G₂ - G₀) ) e^{ λ t } ] + G₂Wait, let me check that.From the earlier expression:G(t) = [ G₁ - C e^{ λ t } G₂ ] / [ 1 - C e^{ λ t } ]Let me factor G₁ in the numerator and denominator:= [ G₁ (1 - (C G₂ / G₁) e^{ λ t }) ] / [ 1 - C e^{ λ t } ]Let me denote D = C G₂ / G₁, then:= G₁ [ 1 - D e^{ λ t } ] / [ 1 - (D G₁ / G₂ ) e^{ λ t } ]But unless D G₁ / G₂ is something specific, this might not help.Alternatively, perhaps I can write it as:G(t) = G₁ - (G₁ - G₂) / [ 1 + ( (G₁ - G₀)/(G₂ - G₀) ) e^{ λ t } ]Wait, let me see:From G(t) = [ G₁ - C e^{ λ t } G₂ ] / [ 1 - C e^{ λ t } ]Let me write this as:G(t) = G₁ [ 1 - (C G₂ / G₁) e^{ λ t } ] / [ 1 - C e^{ λ t } ]Let me denote K = C G₂ / G₁, then:G(t) = G₁ [ 1 - K e^{ λ t } ] / [ 1 - (K G₁ / G₂ ) e^{ λ t } ]But this seems more complicated.Alternatively, perhaps I can rearrange the expression:G(t) = [ G₁ - C G₂ e^{ λ t } ] / [ 1 - C e^{ λ t } ]Let me write this as:G(t) = G₁ + [ - C G₂ e^{ λ t } - G₁ + G₁ ] / [ 1 - C e^{ λ t } ]Wait, that might not help.Alternatively, let me consider the expression:G(t) = [ G₁ - C G₂ e^{ λ t } ] / [ 1 - C e^{ λ t } ]Let me factor out e^{ λ t } in numerator and denominator:= [ e^{ λ t } (G₁ e^{- λ t } - C G₂ ) ] / [ e^{ λ t } (e^{- λ t } - C ) ]Cancel e^{ λ t }:= (G₁ e^{- λ t } - C G₂ ) / (e^{- λ t } - C )Now, let me denote μ = e^{- λ t }, then:G(t) = (G₁ μ - C G₂ ) / (μ - C )= [ G₁ μ - C G₂ ] / (μ - C )Let me factor out C in the numerator:= [ C ( (G₁ / C ) μ - G₂ ) ] / (μ - C )But unless G₁ / C is something specific, this might not help.Alternatively, perhaps I can write it as:G(t) = (G₁ μ - C G₂ ) / (μ - C ) = (G₁ μ - C G₂ ) / (μ - C )Let me perform polynomial division or see if I can express this as a constant plus something over (μ - C ).Let me write the numerator as G₁ μ - C G₂ = A (μ - C ) + BSolve for A and B:G₁ μ - C G₂ = A μ - A C + BEquate coefficients:A = G₁- A C + B = - C G₂So,- G₁ C + B = - C G₂ => B = C (G₁ - G₂ )Therefore,G(t) = [ A (μ - C ) + B ] / (μ - C ) = A + B / (μ - C )= G₁ + [ C (G₁ - G₂ ) ] / (μ - C )But μ = e^{- λ t }, so:G(t) = G₁ + [ C (G₁ - G₂ ) ] / (e^{- λ t } - C )This might be a useful form.Recall that C = (G₁ - G₀)/(G₂ - G₀ )So,G(t) = G₁ + [ (G₁ - G₀)/(G₂ - G₀ ) (G₁ - G₂ ) ] / (e^{- λ t } - (G₁ - G₀)/(G₂ - G₀ ) )Simplify the numerator:(G₁ - G₀)(G₁ - G₂ ) / (G₂ - G₀ )Note that G₁ - G₂ = - (G₂ - G₁ )So,= - (G₁ - G₀)(G₂ - G₁ ) / (G₂ - G₀ )= (G₁ - G₀)(G₁ - G₂ ) / (G₂ - G₀ )Wait, actually, let me compute:(G₁ - G₀)(G₁ - G₂ ) / (G₂ - G₀ )= (G₁ - G₀)(G₁ - G₂ ) / (G₂ - G₀ )= (G₁ - G₀)(G₁ - G₂ ) / -(G₀ - G₂ )= - (G₁ - G₀)(G₁ - G₂ ) / (G₀ - G₂ )But I'm not sure if this helps.Alternatively, perhaps I can leave it as:G(t) = G₁ + [ C (G₁ - G₂ ) ] / (e^{- λ t } - C )where C = (G₁ - G₀)/(G₂ - G₀ )This is a possible expression, but it's still quite involved.Given the time I've spent on this, I think it's acceptable to present the solution in terms of the roots G₁ and G₂, with the constant C determined by the initial condition.So, for part 1, the expression for G(t) is:G(t) = [ G₁ - C e^{ λ t } G₂ ] / [ 1 - C e^{ λ t } ]where G₁ and G₂ are the roots of the quadratic equation k₁ G² - k₁ M G + c I₀ M = 0, and C is given by C = (G₁ - G₀)/(G₂ - G₀ ).Alternatively, since G₁ and G₂ are functions of M, c, I₀, and k₁, we can write them explicitly as:G₁ = [ M + sqrt(M² - 4 c I₀ M / k₁ ) ] / 2G₂ = [ M - sqrt(M² - 4 c I₀ M / k₁ ) ] / 2So, substituting these into the expression for G(t), we have the complete solution.Now, moving on to part 2.The investment inflow I(t) is modeled by:dI/dt = k₂ I(t) (1 - I(t)/N ) + b log(G(t) + 1 )We need to analyze the stability of the equilibrium points and determine the long-term behavior of I(t) when G(t) is as obtained in part 1.First, let's find the equilibrium points of the system. Equilibrium points occur when dI/dt = 0.So,k₂ I (1 - I/N ) + b log(G + 1 ) = 0But G(t) is given by the solution from part 1, which is a function of t. However, in the long term, as t approaches infinity, G(t) may approach an equilibrium value or continue to change. So, to analyze the stability, we need to consider the behavior of I(t) as t approaches infinity, assuming G(t) has reached its long-term behavior.From part 1, depending on the parameters, G(t) could approach one of the equilibrium points G₁ or G₂, or it could grow without bound if I₀ is too large.But since I(t) is a function of G(t), which is itself a function of I₀, which is constant in part 1, we need to consider the interplay between G(t) and I(t).Wait, actually, in part 1, I(t) is constant, I₀, but in part 2, I(t) is a function that depends on G(t). So, the system is actually coupled: G(t) depends on I(t), and I(t) depends on G(t). However, in part 1, we assumed I(t) is constant, but in reality, I(t) is a function that depends on G(t), so the two equations are coupled.But the problem says in part 1, I(t) is a constant I₀, so we can solve for G(t) given I(t)=I₀. Then, in part 2, we have to analyze the system where I(t) is given by its own differential equation, with G(t) as obtained in part 1.Wait, no, actually, the problem says:"Analyze the stability of the equilibrium points of this system and determine the long-term behavior of I(t) when G(t) is as obtained in the first sub-problem."So, in part 2, we need to consider the system where G(t) is given by the solution from part 1, which depends on I₀, and then analyze I(t)'s behavior.But actually, in part 1, I(t) is constant, so in part 2, we need to consider the system where I(t) is no longer constant but follows its own equation, with G(t) as obtained in part 1. However, since G(t) in part 1 depends on I₀, which is a constant, but in part 2, I(t) is a function, not a constant, so perhaps the two equations are coupled, and we need to analyze the system together.Wait, the problem statement is a bit unclear. Let me re-read it."2. The investment inflow, I(t), is modeled by another differential equation: [...] Analyze the stability of the equilibrium points of this system and determine the long-term behavior of I(t) when G(t) is as obtained in the first sub-problem."So, it's saying that in part 2, we have the differential equation for I(t), and we need to analyze its equilibrium points and long-term behavior, assuming that G(t) is as obtained in part 1, i.e., G(t) is given by the solution from part 1, which assumes I(t) is constant I₀.Wait, but if G(t) is given by part 1, which assumes I(t)=I₀, then in part 2, we're treating I(t) as a function that depends on G(t), which is itself a function that depends on I₀. This seems a bit circular.Alternatively, perhaps in part 2, we're to consider the system where both G(t) and I(t) are variables, with G(t) following the equation from part 1, and I(t) following the equation from part 2, making it a system of two coupled differential equations.But the problem says in part 2: "when G(t) is as obtained in the first sub-problem." So, perhaps in part 2, G(t) is given as the solution from part 1, with I(t) constant, and then we analyze I(t)'s behavior given that G(t) is fixed as per part 1.But that would mean that in part 2, G(t) is not a variable but a known function, so the equation for I(t) becomes:dI/dt = k₂ I (1 - I/N ) + b log(G(t) + 1 )Where G(t) is known from part 1.But since G(t) is a function of t, this makes the equation for I(t) a non-autonomous differential equation, which complicates things.Alternatively, perhaps the problem is intended to consider the system where both G(t) and I(t) are variables, and in part 1, we assume I(t) is constant to find G(t), and in part 2, we consider the full system where I(t) is a function of G(t), and analyze the equilibria and stability.Given that, perhaps we should treat the system as two coupled equations:dG/dt = k₁ G (1 - G/M ) - c IdI/dt = k₂ I (1 - I/N ) + b log(G + 1 )And find the equilibrium points of this system, then analyze their stability.But the problem says in part 2: "when G(t) is as obtained in the first sub-problem." So, perhaps in part 2, we're to fix G(t) as the solution from part 1, which assumes I(t)=I₀, and then analyze I(t)'s behavior given that G(t) is fixed.But that would mean that in part 2, G(t) is not a variable but a known function, so the equation for I(t) is:dI/dt = k₂ I (1 - I/N ) + b log(G(t) + 1 )Where G(t) is known from part 1, which depends on I₀.But since G(t) is a function of t, this is a non-autonomous equation, and the analysis would involve finding the behavior of I(t) as t increases, given G(t) is known.Alternatively, perhaps the problem is intended to consider the system where both G and I are variables, and in part 1, we solve for G(t) assuming I is constant, and in part 2, we solve for I(t) assuming G(t) is as found in part 1, which is a function of t.But this would require solving a system where G(t) is known and I(t) is a function of G(t). However, since G(t) is a function of t, and I(t) depends on G(t), it's a non-autonomous system, which is more complex.Alternatively, perhaps the problem is intended to consider the system at equilibrium, where both G and I are constants. So, in part 1, we found G(t) assuming I is constant, and in part 2, we find I(t) assuming G is constant, but that might not capture the full dynamics.Wait, perhaps the problem is intended to have us consider the system where both G and I are variables, and find the equilibrium points where both dG/dt = 0 and dI/dt = 0.So, let's consider that approach.To find equilibrium points, set dG/dt = 0 and dI/dt = 0.From part 1:dG/dt = k₁ G (1 - G/M ) - c I = 0 => c I = k₁ G (1 - G/M )From part 2:dI/dt = k₂ I (1 - I/N ) + b log(G + 1 ) = 0So, we have the system:1. c I = k₁ G (1 - G/M )2. k₂ I (1 - I/N ) + b log(G + 1 ) = 0We need to solve this system for G and I.This is a system of nonlinear equations, and solving it analytically might be difficult. However, we can analyze the possible solutions and their stability.First, let's consider the possible equilibrium points.From equation 1:I = (k₁ / c ) G (1 - G/M )Substitute this into equation 2:k₂ [ (k₁ / c ) G (1 - G/M ) ] (1 - [ (k₁ / c ) G (1 - G/M ) ] / N ) + b log(G + 1 ) = 0This is a complicated equation in G. It's unlikely we can solve this analytically, so we might need to consider the behavior qualitatively or look for possible solutions.Alternatively, we can consider the case where G is at its equilibrium from part 1, i.e., G = G₁ or G = G₂, and see if I can be found accordingly.From part 1, at equilibrium, G = G₁ or G₂, so I = (k₁ / c ) G (1 - G/M )But since at equilibrium, G = G₁ or G₂, which satisfy k₁ G (1 - G/M ) = c I₀, so I = I₀.Wait, but in part 1, I(t) is constant I₀, so in the equilibrium, I = I₀.But in part 2, we have a different equation for I(t), so perhaps the equilibrium points are different.Alternatively, perhaps the equilibrium points are the same as in part 1, but with I(t) adjusted accordingly.This is getting quite involved, and I'm not sure if I can proceed further without more information.Alternatively, perhaps we can linearize the system around the equilibrium points and determine their stability.Let me denote the equilibrium points as (G*, I*), where:c I* = k₁ G* (1 - G*/M )andk₂ I* (1 - I*/N ) + b log(G* + 1 ) = 0To analyze stability, we can compute the Jacobian matrix of the system at (G*, I*) and determine the eigenvalues.The Jacobian matrix J is:[ ∂(dG/dt)/∂G , ∂(dG/dt)/∂I ][ ∂(dI/dt)/∂G , ∂(dI/dt)/∂I ]Compute each partial derivative:∂(dG/dt)/∂G = k₁ (1 - G/M ) - k₁ G (1/M ) = k₁ (1 - 2 G/M )∂(dG/dt)/∂I = -c∂(dI/dt)/∂G = b / (G + 1 )∂(dI/dt)/∂I = k₂ (1 - 2 I/N )So, the Jacobian matrix is:[ k₁ (1 - 2 G/M ) , -c ][ b / (G + 1 ) , k₂ (1 - 2 I/N ) ]At the equilibrium point (G*, I*), we substitute G = G*, I = I*.The eigenvalues of this matrix determine the stability. If both eigenvalues have negative real parts, the equilibrium is stable; if any eigenvalue has a positive real part, it's unstable.But without knowing the specific values of G* and I*, it's difficult to compute the eigenvalues. However, we can analyze the signs.Alternatively, perhaps we can consider specific cases or make assumptions about the parameters to simplify the analysis.But given the complexity, I think it's beyond the scope of this problem to find an explicit solution for I(t). Instead, we can discuss the possible long-term behaviors based on the system's dynamics.Given that, in part 1, G(t) approaches an equilibrium value (assuming I₀ is such that real equilibria exist), then in part 2, with G(t) approaching G*, I(t) will be influenced by the term b log(G* + 1 ). So, the equation for I(t) becomes:dI/dt = k₂ I (1 - I/N ) + b log(G* + 1 )This is a logistic equation for I(t) with an additional constant term. The behavior of I(t) will depend on the value of b log(G* + 1 ).If b log(G* + 1 ) is positive, it acts as a source term, potentially shifting the equilibrium point. If it's negative, it acts as a sink.The equilibrium points for I(t) when G(t) is fixed at G* are found by setting dI/dt = 0:k₂ I (1 - I/N ) + b log(G* + 1 ) = 0This is a quadratic equation in I:k₂ I - (k₂ / N ) I² + b log(G* + 1 ) = 0Multiply through by -N:(k₂ / N ) I² - k₂ I - b N log(G* + 1 ) = 0So,I² - N I - (b N² / k₂ ) log(G* + 1 ) = 0The solutions are:I = [ N ± sqrt(N² + 4 (b N² / k₂ ) log(G* + 1 )) ] / 2So, the equilibrium points for I(t) are:I* = [ N ± sqrt(N² + 4 (b N² / k₂ ) log(G* + 1 )) ] / 2The stability of these points depends on the derivative of dI/dt with respect to I at these points.The derivative is:d(dI/dt)/dI = k₂ (1 - 2 I/N )At I = I*, the derivative is:k₂ (1 - 2 I*/N )If this is negative, the equilibrium is stable; if positive, unstable.Let me compute this for each equilibrium.First, for I* = [ N + sqrt(N² + 4 (b N² / k₂ ) log(G* + 1 )) ] / 2Then,1 - 2 I*/N = 1 - 2 [ N + sqrt(N² + 4 (b N² / k₂ ) log(G* + 1 )) ] / (2 N )= 1 - [ N + sqrt(N² + 4 (b N² / k₂ ) log(G* + 1 )) ] / N= 1 - 1 - sqrt(N² + 4 (b N² / k₂ ) log(G* + 1 )) / N= - sqrt(1 + 4 (b / k₂ ) log(G* + 1 )) Which is negative, so this equilibrium is stable.For I* = [ N - sqrt(N² + 4 (b N² / k₂ ) log(G* + 1 )) ] / 2Similarly,1 - 2 I*/N = 1 - 2 [ N - sqrt(N² + 4 (b N² / k₂ ) log(G* + 1 )) ] / (2 N )= 1 - [ N - sqrt(N² + 4 (b N² / k₂ ) log(G* + 1 )) ] / N= 1 - 1 + sqrt(N² + 4 (b N² / k₂ ) log(G* + 1 )) / N= sqrt(1 + 4 (b / k₂ ) log(G* + 1 )) Which is positive, so this equilibrium is unstable.Therefore, the system will approach the stable equilibrium I* = [ N + sqrt(N² + 4 (b N² / k₂ ) log(G* + 1 )) ] / 2 as t approaches infinity.But wait, this is under the assumption that G(t) has reached its equilibrium G*. However, in reality, G(t) is also changing over time, so the analysis is more complex.Alternatively, if we consider that G(t) approaches G* as t increases, then I(t) will approach the stable equilibrium I* as above.Therefore, the long-term behavior of I(t) is that it approaches the stable equilibrium point given by:I* = [ N + sqrt(N² + 4 (b N² / k₂ ) log(G* + 1 )) ] / 2where G* is the equilibrium value of G(t) from part 1.But since G* depends on I₀, which is a constant in part 1, and in part 2, I(t) is a function, this creates a feedback loop. However, given the problem's structure, it's likely that the long-term behavior of I(t) is to approach this stable equilibrium.In summary, for part 2, the system has equilibrium points for I(t) given by the solutions to the quadratic equation, with the stable equilibrium being the larger root, and the unstable one being the smaller root. Therefore, the long-term behavior of I(t) is to approach the stable equilibrium I*.However, this is under the assumption that G(t) has reached its equilibrium G*. If G(t) is still changing, the analysis becomes more involved, but given the problem's context, it's reasonable to assume that G(t) approaches G*, and then I(t) approaches I*.Therefore, the long-term behavior of I(t) is convergence to the stable equilibrium point I*.So, to recap:1. The expression for G(t) is a solution to the Riccati equation, which can be written in terms of the roots G₁ and G₂, with a constant determined by the initial condition.2. The system for I(t) has equilibrium points which depend on G*, and the stable equilibrium for I(t) is the larger root of the quadratic equation derived from setting dI/dt = 0.Given the complexity of the expressions, I think it's acceptable to present the answers in terms of the roots and the derived equilibrium points.</think>

🧠:<think>Alright, so I've got these two math problems to solve, both related to a video game scenario. Let me take them one at a time.Starting with the first problem: It involves a function ( G(t) = e^{kt} sin(omega t) ), where ( k ) and ( omega ) are positive constants. I need to find the time ( t ) at which the player's game sense reaches its first local maximum after ( t = 0 ). Hmm, okay. So, I remember that to find local maxima or minima of a function, we take its derivative and set it equal to zero. That should give us the critical points, and then we can determine which ones are maxima or minima.So, let's compute the derivative of ( G(t) ). Since ( G(t) ) is the product of two functions, ( e^{kt} ) and ( sin(omega t) ), I'll need to use the product rule. The product rule states that ( (uv)' = u'v + uv' ).Let me denote ( u = e^{kt} ) and ( v = sin(omega t) ). Then, ( u' = k e^{kt} ) and ( v' = omega cos(omega t) ).Putting it all together, the derivative ( G'(t) ) is:( G'(t) = k e^{kt} sin(omega t) + e^{kt} omega cos(omega t) )We can factor out ( e^{kt} ) since it's a common term:( G'(t) = e^{kt} [k sin(omega t) + omega cos(omega t)] )To find the critical points, we set ( G'(t) = 0 ). Since ( e^{kt} ) is always positive for any real ( t ), we can divide both sides by ( e^{kt} ) without changing the equality:( k sin(omega t) + omega cos(omega t) = 0 )So, the equation simplifies to:( k sin(omega t) + omega cos(omega t) = 0 )Let me rearrange this equation:( k sin(omega t) = -omega cos(omega t) )Divide both sides by ( cos(omega t) ) (assuming ( cos(omega t) neq 0 )):( k tan(omega t) = -omega )So,( tan(omega t) = -frac{omega}{k} )Hmm, okay. So, we need to solve for ( t ) in this equation. Let's denote ( theta = omega t ), so the equation becomes:( tan(theta) = -frac{omega}{k} )We can write this as:( theta = arctanleft(-frac{omega}{k}right) + npi ), where ( n ) is an integer.But since we're looking for the first local maximum after ( t = 0 ), we need the smallest positive ( t ) such that this equation holds. Let's analyze the behavior of ( G(t) ) around ( t = 0 ).At ( t = 0 ), ( G(t) = e^{0} sin(0) = 0 ). So, the function starts at zero. As ( t ) increases, ( e^{kt} ) grows exponentially, and ( sin(omega t) ) oscillates between -1 and 1. So, the product ( G(t) ) will oscillate with increasing amplitude.To find the first local maximum, we can consider the first positive solution to ( G'(t) = 0 ) where the function changes from increasing to decreasing. So, we need the first positive ( t ) where ( tan(omega t) = -frac{omega}{k} ).But wait, ( tan(theta) = -frac{omega}{k} ). Since ( omega ) and ( k ) are positive constants, the right-hand side is negative. So, ( tan(theta) ) is negative. The tangent function is negative in the second and fourth quadrants.But since ( theta = omega t ) and ( t > 0 ), ( theta ) is positive. So, the first solution where ( tan(theta) ) is negative would be in the second quadrant, i.e., ( pi/2 < theta < pi ).So, the principal value of ( arctan(-frac{omega}{k}) ) is negative, but we can add ( pi ) to get the angle in the second quadrant.Thus, the first positive solution is:( theta = pi - arctanleft(frac{omega}{k}right) )Therefore, ( omega t = pi - arctanleft(frac{omega}{k}right) )So, solving for ( t ):( t = frac{1}{omega} left( pi - arctanleft( frac{omega}{k} right) right) )Hmm, let me check if this makes sense. Let's consider the case when ( k ) is very large compared to ( omega ). Then, ( arctan(frac{omega}{k}) ) approaches zero, so ( t ) approaches ( pi / omega ). That seems reasonable because if the adaptability is high, the function might peak earlier.Alternatively, if ( omega ) is very large compared to ( k ), then ( arctan(frac{omega}{k}) ) approaches ( pi/2 ), so ( t ) approaches ( (pi - pi/2)/omega = pi/(2omega) ). That also makes sense because a higher frequency would lead to a peak sooner.Wait, but is this the first local maximum? Let me think. The function ( G(t) ) starts at zero, increases, reaches a peak, then decreases, and so on. So, the first critical point after ( t = 0 ) is a local maximum.But let me double-check by considering the second derivative or the sign changes. Alternatively, I can test values around the critical point.Suppose ( t ) is slightly less than the critical point ( t_c ). Then, ( G'(t) ) would be positive because the function is increasing towards the peak. After ( t_c ), ( G'(t) ) becomes negative as the function starts decreasing. Therefore, ( t_c ) is indeed a local maximum.So, the time ( t ) at which the first local maximum occurs is ( t = frac{1}{omega} left( pi - arctanleft( frac{omega}{k} right) right) ).But let me see if I can express this in a different form. Since ( arctan(x) + arctan(1/x) = pi/2 ) for ( x > 0 ), maybe I can rewrite it.Let me denote ( alpha = arctanleft( frac{omega}{k} right) ). Then, ( arctanleft( frac{k}{omega} right) = pi/2 - alpha ).But in our case, we have ( pi - alpha ). Hmm, not sure if that helps. Maybe it's fine as it is.So, I think that's the answer for the first part.Moving on to the second problem: We have a probability function ( P(x, y) = e^{-(x^2 + y^2)} ). We need to find the critical points and determine whether each is a local maximum, local minimum, or saddle point.Alright, critical points occur where the gradient is zero, i.e., where both partial derivatives with respect to ( x ) and ( y ) are zero.Let me compute the partial derivatives.First, the partial derivative with respect to ( x ):( frac{partial P}{partial x} = frac{partial}{partial x} e^{-(x^2 + y^2)} = e^{-(x^2 + y^2)} cdot (-2x) = -2x e^{-(x^2 + y^2)} )Similarly, the partial derivative with respect to ( y ):( frac{partial P}{partial y} = frac{partial}{partial y} e^{-(x^2 + y^2)} = e^{-(x^2 + y^2)} cdot (-2y) = -2y e^{-(x^2 + y^2)} )So, to find critical points, set both partial derivatives equal to zero:1. ( -2x e^{-(x^2 + y^2)} = 0 )2. ( -2y e^{-(x^2 + y^2)} = 0 )Since ( e^{-(x^2 + y^2)} ) is never zero for any real ( x ) and ( y ), we can divide both sides by it, leading to:1. ( -2x = 0 ) => ( x = 0 )2. ( -2y = 0 ) => ( y = 0 )So, the only critical point is at ( (0, 0) ).Now, to determine the nature of this critical point, we can use the second derivative test. For functions of two variables, we compute the Hessian matrix, which consists of the second partial derivatives.Compute the second partial derivatives:First, ( frac{partial^2 P}{partial x^2} ):( frac{partial}{partial x} (-2x e^{-(x^2 + y^2)}) = -2 e^{-(x^2 + y^2)} + (-2x)(-2x) e^{-(x^2 + y^2)} = -2 e^{-(x^2 + y^2)} + 4x^2 e^{-(x^2 + y^2)} )Simplify:( frac{partial^2 P}{partial x^2} = (-2 + 4x^2) e^{-(x^2 + y^2)} )Similarly, ( frac{partial^2 P}{partial y^2} ):( frac{partial}{partial y} (-2y e^{-(x^2 + y^2)}) = -2 e^{-(x^2 + y^2)} + (-2y)(-2y) e^{-(x^2 + y^2)} = -2 e^{-(x^2 + y^2)} + 4y^2 e^{-(x^2 + y^2)} )Simplify:( frac{partial^2 P}{partial y^2} = (-2 + 4y^2) e^{-(x^2 + y^2)} )Now, the mixed partial derivatives ( frac{partial^2 P}{partial x partial y} ) and ( frac{partial^2 P}{partial y partial x} ). Since the function is smooth, these should be equal.Compute ( frac{partial^2 P}{partial x partial y} ):First, take the partial derivative of ( frac{partial P}{partial x} ) with respect to ( y ):( frac{partial}{partial y} (-2x e^{-(x^2 + y^2)}) = (-2x)(-2y) e^{-(x^2 + y^2)} = 4xy e^{-(x^2 + y^2)} )So, ( frac{partial^2 P}{partial x partial y} = 4xy e^{-(x^2 + y^2)} )Similarly, ( frac{partial^2 P}{partial y partial x} = 4xy e^{-(x^2 + y^2)} )So, the Hessian matrix ( H ) at the critical point ( (0, 0) ) is:( H = begin{bmatrix}frac{partial^2 P}{partial x^2} & frac{partial^2 P}{partial x partial y} frac{partial^2 P}{partial y partial x} & frac{partial^2 P}{partial y^2}end{bmatrix} )Evaluating at ( (0, 0) ):( frac{partial^2 P}{partial x^2} = (-2 + 0) e^{0} = -2 )( frac{partial^2 P}{partial y^2} = (-2 + 0) e^{0} = -2 )( frac{partial^2 P}{partial x partial y} = 0 )So,( H = begin{bmatrix}-2 & 0 0 & -2end{bmatrix} )The determinant of the Hessian is ( (-2)(-2) - (0)^2 = 4 ), which is positive. Also, the leading principal minor (the top-left element) is ( -2 ), which is negative. In the second derivative test for functions of two variables, if the determinant is positive and the leading principal minor is negative, the critical point is a local maximum.Therefore, the critical point at ( (0, 0) ) is a local maximum.Wait, but let me think again. The function ( P(x, y) = e^{-(x^2 + y^2)} ) is a Gaussian function, which is a bell-shaped curve in two dimensions. It has a single peak at the origin, which is indeed a local (and global) maximum. So, this makes sense.So, summarizing, the only critical point is at ( (0, 0) ), and it's a local maximum.I think that's all for the second problem.Final Answer1. The first local maximum occurs at ( t = boxed{dfrac{pi - arctanleft(dfrac{omega}{k}right)}{omega}} ).2. The critical point is at ( (0, 0) ) and it is a boxed{text{local maximum}}.</think>