🧠:<think>Alright, so I've got this problem about an actress modeling her career impact with a mathematical function. It's split into two parts: finding critical points and classifying them, and then calculating the total impact over a decade. Let me tackle each part step by step.Starting with part 1: The function given is ( I(t) = A e^{kt} sin(omega t + phi) + B ). I need to find the critical points, which means I have to take the derivative of I(t) with respect to t and set it equal to zero. Then, I can classify those points as maxima, minima, or points of inflection.First, let me recall that critical points occur where the first derivative is zero or undefined. Since this function is a combination of exponential and sinusoidal functions, which are both smooth and differentiable everywhere, the critical points will only occur where the derivative is zero.So, let's compute the derivative ( I'(t) ). The function has two main parts: ( A e^{kt} sin(omega t + phi) ) and the constant B. The derivative of B is zero, so I can focus on the first part.Using the product rule for differentiation, since we have a product of ( e^{kt} ) and ( sin(omega t + phi) ). The product rule states that ( (uv)' = u'v + uv' ).Let me set ( u = A e^{kt} ) and ( v = sin(omega t + phi) ). Then, ( u' = A k e^{kt} ) and ( v' = omega cos(omega t + phi) ).Putting it all together, the derivative is:( I'(t) = A k e^{kt} sin(omega t + phi) + A e^{kt} omega cos(omega t + phi) )I can factor out ( A e^{kt} ) from both terms:( I'(t) = A e^{kt} [k sin(omega t + phi) + omega cos(omega t + phi)] )To find critical points, set ( I'(t) = 0 ). Since ( A e^{kt} ) is always positive (as A and k are positive constants and exponential function is always positive), the equation simplifies to:( k sin(omega t + phi) + omega cos(omega t + phi) = 0 )Let me write this as:( k sin(theta) + omega cos(theta) = 0 ), where ( theta = omega t + phi )This is a trigonometric equation. I can rewrite it in terms of a single sine or cosine function. Let's use the identity that ( a sin theta + b cos theta = R sin(theta + delta) ), where ( R = sqrt{a^2 + b^2} ) and ( delta = arctanleft(frac{b}{a}right) ) or something like that.Wait, actually, the identity is ( a sin theta + b cos theta = R sin(theta + delta) ), where ( R = sqrt{a^2 + b^2} ) and ( delta = arctanleft(frac{b}{a}right) ). Alternatively, it can also be written as ( R cos(theta - delta) ), depending on the phase shift.Let me compute R:( R = sqrt{k^2 + omega^2} )And the angle ( delta ) can be found by:( tan delta = frac{omega}{k} ), so ( delta = arctanleft(frac{omega}{k}right) )Therefore, the equation becomes:( R sin(theta + delta) = 0 )Which implies:( sin(theta + delta) = 0 )So, ( theta + delta = npi ), where n is an integer.Substituting back ( theta = omega t + phi ):( omega t + phi + delta = npi )Solving for t:( t = frac{npi - phi - delta}{omega} )So, these are the critical points. Now, to classify them as maxima, minima, or points of inflection, I need to look at the second derivative or use the first derivative test.But since this is a product of an exponential function and a sinusoidal function, the critical points will alternate between maxima and minima, given the periodic nature of the sine function. However, because the exponential function is always increasing (since k is positive), the amplitude of these critical points will increase over time.Alternatively, I can compute the second derivative ( I''(t) ) and evaluate it at the critical points.Let me compute ( I''(t) ). Starting from ( I'(t) = A e^{kt} [k sin(theta) + omega cos(theta)] ), where ( theta = omega t + phi ).Differentiating again, using the product rule:( I''(t) = A [k e^{kt} [k sin(theta) + omega cos(theta)] + e^{kt} [k^2 cos(theta) - omega^2 sin(theta)] ] )Wait, let me do this step by step.First, ( I'(t) = A e^{kt} [k sin(theta) + omega cos(theta)] )So, ( I''(t) = A [ d/dt (e^{kt}) [k sin(theta) + omega cos(theta)] + e^{kt} d/dt [k sin(theta) + omega cos(theta)] ] )Compute each part:1. ( d/dt (e^{kt}) = k e^{kt} )2. ( d/dt [k sin(theta) + omega cos(theta)] = k omega cos(theta) - omega^2 sin(theta) )Putting it together:( I''(t) = A [k e^{kt} [k sin(theta) + omega cos(theta)] + e^{kt} [k omega cos(theta) - omega^2 sin(theta)] ] )Factor out ( e^{kt} ):( I''(t) = A e^{kt} [k(k sin(theta) + omega cos(theta)) + (k omega cos(theta) - omega^2 sin(theta))] )Simplify inside the brackets:First term: ( k^2 sin(theta) + k omega cos(theta) )Second term: ( k omega cos(theta) - omega^2 sin(theta) )Combine like terms:- ( sin(theta) ): ( k^2 - omega^2 )- ( cos(theta) ): ( k omega + k omega = 2 k omega )So, overall:( I''(t) = A e^{kt} [ (k^2 - omega^2) sin(theta) + 2 k omega cos(theta) ] )Now, at the critical points, we have ( k sin(theta) + omega cos(theta) = 0 ). Let's denote this as equation (1).From equation (1), we can express ( sin(theta) = - frac{omega}{k} cos(theta) ).Let me substitute this into the expression for ( I''(t) ):( I''(t) = A e^{kt} [ (k^2 - omega^2)( - frac{omega}{k} cos(theta) ) + 2 k omega cos(theta) ] )Simplify:First term: ( (k^2 - omega^2)( - frac{omega}{k} ) cos(theta) = - frac{omega}{k} (k^2 - omega^2) cos(theta) )Second term: ( 2 k omega cos(theta) )Combine them:( [ - frac{omega}{k} (k^2 - omega^2) + 2 k omega ] cos(theta) )Factor out ( omega cos(theta) ):( omega cos(theta) [ - frac{(k^2 - omega^2)}{k} + 2 k ] )Simplify the expression inside the brackets:Let me compute ( - frac{(k^2 - omega^2)}{k} + 2 k ):= ( - frac{k^2}{k} + frac{omega^2}{k} + 2k )= ( -k + frac{omega^2}{k} + 2k )= ( ( -k + 2k ) + frac{omega^2}{k} )= ( k + frac{omega^2}{k} )= ( frac{k^2 + omega^2}{k} )So, putting it all together:( I''(t) = A e^{kt} cdot omega cos(theta) cdot frac{k^2 + omega^2}{k} )Simplify:( I''(t) = A e^{kt} cdot frac{omega (k^2 + omega^2)}{k} cos(theta) )Now, since ( A, e^{kt}, omega, k, ) and ( (k^2 + omega^2) ) are all positive constants, the sign of ( I''(t) ) depends on ( cos(theta) ).Recall that ( theta = omega t + phi ). At the critical points, we have ( sin(theta) = - frac{omega}{k} cos(theta) ). Let's see if we can find the sign of ( cos(theta) ) at these points.From equation (1):( k sin(theta) + omega cos(theta) = 0 )=> ( sin(theta) = - frac{omega}{k} cos(theta) )Square both sides:( sin^2(theta) = frac{omega^2}{k^2} cos^2(theta) )Using the identity ( sin^2(theta) + cos^2(theta) = 1 ):( frac{omega^2}{k^2} cos^2(theta) + cos^2(theta) = 1 )Factor out ( cos^2(theta) ):( cos^2(theta) left( frac{omega^2}{k^2} + 1 right) = 1 )=> ( cos^2(theta) = frac{1}{1 + frac{omega^2}{k^2}} = frac{k^2}{k^2 + omega^2} )Thus, ( cos(theta) = pm frac{k}{sqrt{k^2 + omega^2}} )So, ( cos(theta) ) can be positive or negative. Therefore, the sign of ( I''(t) ) depends on the sign of ( cos(theta) ).If ( cos(theta) > 0 ), then ( I''(t) > 0 ), which means the critical point is a local minimum.If ( cos(theta) < 0 ), then ( I''(t) < 0 ), which means the critical point is a local maximum.Therefore, the critical points alternate between maxima and minima depending on the value of ( cos(theta) ) at those points.So, summarizing part 1:The critical points occur at ( t = frac{npi - phi - delta}{omega} ), where ( delta = arctanleft(frac{omega}{k}right) ) and n is an integer. These points are either local maxima or minima depending on whether ( cos(theta) ) is negative or positive at those points, respectively.Moving on to part 2: The actress wants to calculate the total impact over a decade, i.e., from t=0 to t=10. So, we need to compute the definite integral ( int_{0}^{10} I(t) dt ).The function is ( I(t) = A e^{kt} sin(omega t + phi) + B ). So, the integral becomes:( int_{0}^{10} [A e^{kt} sin(omega t + phi) + B] dt = A int_{0}^{10} e^{kt} sin(omega t + phi) dt + B int_{0}^{10} dt )Let me compute each integral separately.First, the integral of B dt from 0 to 10 is straightforward:( B int_{0}^{10} dt = B [t]_{0}^{10} = B(10 - 0) = 10B )Now, the more complex part is the integral ( int e^{kt} sin(omega t + phi) dt ). This is a standard integral that can be solved using integration by parts or by using a formula.I recall that the integral of ( e^{at} sin(bt + c) dt ) is:( frac{e^{at}}{a^2 + b^2} (a sin(bt + c) - b cos(bt + c)) + C )Similarly, the integral of ( e^{at} cos(bt + c) dt ) is:( frac{e^{at}}{a^2 + b^2} (a cos(bt + c) + b sin(bt + c)) + C )So, applying this formula to our integral where a = k and b = ω, c = φ.Thus,( int e^{kt} sin(omega t + phi) dt = frac{e^{kt}}{k^2 + omega^2} [k sin(omega t + phi) - omega cos(omega t + phi)] + C )Therefore, the definite integral from 0 to 10 is:( A left[ frac{e^{kt}}{k^2 + omega^2} [k sin(omega t + phi) - omega cos(omega t + phi)] right]_{0}^{10} )Let me compute this:First, evaluate at t=10:( frac{e^{10k}}{k^2 + omega^2} [k sin(10omega + phi) - omega cos(10omega + phi)] )Then, evaluate at t=0:( frac{e^{0}}{k^2 + omega^2} [k sin(phi) - omega cos(phi)] = frac{1}{k^2 + omega^2} [k sin(phi) - omega cos(phi)] )Subtracting the lower limit from the upper limit:( frac{A}{k^2 + omega^2} left[ e^{10k} [k sin(10omega + phi) - omega cos(10omega + phi)] - [k sin(phi) - omega cos(phi)] right] )So, combining both integrals, the total impact is:( frac{A}{k^2 + omega^2} left[ e^{10k} [k sin(10omega + phi) - omega cos(10omega + phi)] - [k sin(phi) - omega cos(phi)] right] + 10B )Now, discussing the conditions under which the integral might not converge. Since we're integrating over a finite interval [0,10], the integral will always converge because the function ( I(t) ) is continuous on this interval. However, if we were integrating over an infinite interval, say from 0 to infinity, the exponential term ( e^{kt} ) would cause the integral to diverge if k > 0, which it is. But since we're only integrating over a decade (10 years), the integral is finite.In terms of the actress's career volatility, the exponential growth term ( e^{kt} ) suggests that her impact is increasing over time. The sinusoidal term adds periodic fluctuations, which could represent varying levels of success or creativity over time. The integral's result shows that the total impact is a combination of her steady growth (captured by the exponential term) and her periodic creative bursts (captured by the sine and cosine terms). If k is large, the exponential term dominates, making the total impact heavily weighted towards the end of the decade. If ω is large, the fluctuations are more rapid, potentially leading to more variability in her impact over the decade.So, summarizing part 2:The total impact over a decade is given by the integral above, which combines both the exponential growth and sinusoidal fluctuations. Since the interval is finite, the integral always converges. The convergence is guaranteed here, but in an infinite interval, the exponential term would cause divergence, indicating that her impact would become unbounded over time, which could relate to her career becoming increasingly volatile or impactful as time goes on.Final Answer1. The critical points occur at ( t = frac{npi - phi - arctanleft(frac{omega}{k}right)}{omega} ) for integer ( n ), and they are classified as local maxima or minima depending on the sign of ( cos(omega t + phi) ) at those points.2. The total impact over a decade is ( frac{A}{k^2 + omega^2} left[ e^{10k} left(k sin(10omega + phi) - omega cos(10omega + phi)right) - left(k sinphi - omega cosphiright) right] + 10B ). The integral converges for all positive constants ( A, k, omega, phi, ) and ( B ) over the interval [0, 10].The final answers are:1. Critical points are at ( boxed{t = frac{npi - phi - arctanleft(frac{omega}{k}right)}{omega}} ) and are maxima or minima based on ( cos(omega t + phi) ).2. Total impact is ( boxed{frac{A}{k^2 + omega^2} left[ e^{10k} left(k sin(10omega + phi) - omega cos(10omega + phi)right) - left(k sinphi - omega cosphiright) right] + 10B} ).</think>

🧠:<think>Alright, so I have this problem where Professor Kwame Mensah is looking at the impact of a new math curriculum in five Sub-Saharan African countries. He's collected data from 100 schools in each country, making a total of 500 schools. The main variables are the average math exam scores (S_i), and some socio-economic factors: average household income (I_i), access to educational resources (R_i), and teacher-student ratio (T_i). First, he wants to do a multivariate regression analysis to see how these socio-economic indicators relate to the student performance. Then, there's an optimization problem where he wants to allocate additional resources to maximize the overall increase in scores, given a budget constraint.Starting with the statistical analysis. I remember that multivariate regression models the relationship between a dependent variable and multiple independent variables. In this case, S_i is the dependent variable, and I_i, R_i, T_i are the independent variables. So, the general form of a multiple regression model is:S_i = β0 + β1*I_i + β2*R_i + β3*T_i + ε_iWhere β0 is the intercept, β1, β2, β3 are the coefficients for each independent variable, and ε_i is the error term.To identify the coefficients, we need to use a method like ordinary least squares (OLS). OLS minimizes the sum of squared residuals, which are the differences between the observed and predicted values of S_i.But wait, do I need to consider any assumptions here? Yeah, OLS assumes linearity, independence of errors, homoscedasticity, no multicollinearity, and normality of errors. I should check if these assumptions hold, but since I don't have the actual data, I can't perform those checks right now. Maybe I can just proceed with the model as is.So, the model is:S_i = β0 + β1*I_i + β2*R_i + β3*T_i + ε_iNow, moving on to the optimization problem. He wants to allocate additional resources A_i to each school to maximize the total increase in scores, given a budget B.The increase in score is given by ΔS_i = α log(1 + β A_i). So, the total increase is the sum of ΔS_i for all schools. The objective is to maximize this sum subject to the constraint that the total allocated resources don't exceed B.Mathematically, this can be written as:Maximize Σ (α log(1 + β A_i)) for i=1 to 500Subject to Σ A_i ≤ B and A_i ≥ 0This looks like a constrained optimization problem. I think I can use the method of Lagrange multipliers here. Let me recall how that works.We can set up the Lagrangian function:L = Σ [α log(1 + β A_i)] - λ (Σ A_i - B)Where λ is the Lagrange multiplier.To find the maximum, we take the partial derivatives of L with respect to each A_i and set them equal to zero.Partial derivative of L with respect to A_i:dL/dA_i = (α β) / (1 + β A_i) - λ = 0Solving for A_i:(α β) / (1 + β A_i) = λ=> 1 + β A_i = (α β)/λ=> A_i = [(α β)/λ - 1] / βSimplify:A_i = (α / λ) - (1 / β)Hmm, but this seems a bit off. Let me double-check the derivative.Yes, derivative of log(1 + β A_i) with respect to A_i is β / (1 + β A_i). So, multiplied by α, it's (α β)/(1 + β A_i). So, the derivative is correct.So, setting the derivative equal to λ gives:(α β)/(1 + β A_i) = λ=> 1 + β A_i = (α β)/λ=> A_i = (α / λ) - (1 / β)Wait, but A_i must be non-negative. So, (α / λ) must be greater than or equal to (1 / β). Otherwise, A_i would be negative, which isn't allowed.So, we have A_i = (α / λ - 1 / β). But since λ is a multiplier, it's determined by the constraint.But how do we find λ? We can use the budget constraint.Σ A_i = BSubstituting A_i from above:Σ [(α / λ - 1 / β)] = BBut wait, this would be 500*(α / λ - 1 / β) = BBecause each A_i is the same? Wait, is that the case? If all A_i are equal, then yes, but in reality, the optimal allocation might depend on the parameters α and β, which are the same for all schools, right?Wait, hold on. The function f(A_i) is the same for all schools, so α and β are constants across all schools. Therefore, the optimal A_i would be the same for all schools, given the same parameters.So, if all A_i are equal, then each A_i = B / 500.But wait, let me think again. The function is concave because the second derivative is negative. So, the marginal increase in score decreases as A_i increases. Therefore, the optimal allocation would spread the resources equally across all schools because the marginal benefit is the same for each school.Wait, no. Wait, the marginal benefit is (α β)/(1 + β A_i). If we set this equal across all schools, then A_i would be the same for all schools. So, yes, equal allocation.Therefore, the optimal allocation is to give each school an equal amount of resources, A_i = B / 500.But let me verify this.Suppose we have two schools, school 1 and school 2. The marginal benefit for school 1 is (α β)/(1 + β A1), and for school 2 it's (α β)/(1 + β A2). To maximize the total, we should allocate resources where the marginal benefit is highest. But since the marginal benefit is the same for all schools (because the function is identical), we can allocate equally.Alternatively, if the functions were different, we might allocate more to schools with higher marginal benefits. But here, since they're the same, equal allocation is optimal.So, in conclusion, the optimal allocation is A_i = B / 500 for each school.Wait, but let me think about the Lagrangian solution again. We had A_i = (α / λ) - (1 / β). If all A_i are equal, then (α / λ) - (1 / β) must be equal for all i, which is consistent. So, the solution is consistent with equal allocation.Therefore, the optimal allocation is to distribute the budget equally among all 500 schools.But let me think if there's another way to approach this. Maybe using calculus of variations or something else, but I think the Lagrangian method suffices here.So, summarizing:1. The regression model is S_i = β0 + β1*I_i + β2*R_i + β3*T_i + ε_i.2. The optimal allocation is A_i = B / 500 for each school.Wait, but in the Lagrangian, we had A_i = (α / λ) - (1 / β). So, if we set all A_i equal, then:A = (α / λ) - (1 / β)And total A_i = 500*A = BSo, 500*(α / λ - 1 / β) = BSolving for λ:500*(α / λ) - 500 / β = B=> 500α / λ = B + 500 / β=> λ = 500α / (B + 500 / β)Therefore, A = (α / λ) - (1 / β) = [α / (500α / (B + 500 / β))] - (1 / β) = (B + 500 / β)/500 - 1 / β = (B / 500) + (1 / β) - (1 / β) = B / 500So, yes, A = B / 500.Therefore, the optimal allocation is to give each school an equal share of the budget.So, that's the conclusion.Final Answer1. The regression model is ( S_i = beta_0 + beta_1 I_i + beta_2 R_i + beta_3 T_i + epsilon_i ), where the coefficients ( beta_0, beta_1, beta_2, beta_3 ) are estimated using ordinary least squares.2. The optimal allocation of resources is ( A_i = frac{B}{500} ) for each school, resulting in the maximum total increase in scores. Thus, the optimal allocation is boxed{A_i = frac{B}{500}}.</think>

🧠:<think>Alright, so I have this problem about a poet's feminist aunt who used group theory to analyze gender roles. The group in question is isomorphic to the dihedral group ( D_{12} ), which is the symmetry group of a regular 12-gon. The first part asks me to determine the number of distinct subgroups of ( G ) and describe their structure in terms of orders and possible isomorphisms with well-known groups. The second part is about a 4-dimensional hypercube projected into 3D, and I need to find the number of distinct planes formed by three projected vertices, relating this to the symmetries of ( G ).Starting with the first part: ( G cong D_{12} ). I remember that dihedral groups have a specific structure. ( D_n ) is the group of symmetries of a regular n-gon, including rotations and reflections. The order of ( D_n ) is ( 2n ), so ( D_{12} ) has order 24. That means ( |G| = 24 ).To find the number of distinct subgroups, I need to recall the subgroup structure of dihedral groups. I think dihedral groups have subgroups of two types: cyclic subgroups and dihedral subgroups. Specifically, for each divisor ( d ) of ( n ), there are cyclic subgroups of order ( d ) and dihedral subgroups of order ( 2d ).So, for ( D_{12} ), ( n = 12 ). The divisors of 12 are 1, 2, 3, 4, 6, 12. For each divisor ( d ), we have a cyclic subgroup of order ( d ) and a dihedral subgroup of order ( 2d ), except when ( d = 1 ), where the dihedral subgroup would be trivial.Let me list them:1. For ( d = 1 ): Cyclic subgroup of order 1 (trivial group). There's only one such subgroup.2. For ( d = 2 ): Cyclic subgroup of order 2 and dihedral subgroup of order 4.3. For ( d = 3 ): Cyclic subgroup of order 3 and dihedral subgroup of order 6.4. For ( d = 4 ): Cyclic subgroup of order 4 and dihedral subgroup of order 8.5. For ( d = 6 ): Cyclic subgroup of order 6 and dihedral subgroup of order 12.6. For ( d = 12 ): Cyclic subgroup of order 12 and dihedral subgroup of order 24, which is the entire group itself.Wait, but hold on. For each ( d ), how many subgroups do we get? For cyclic subgroups, each divisor ( d ) gives exactly one cyclic subgroup of order ( d ). Similarly, for dihedral subgroups, each divisor ( d ) gives exactly one dihedral subgroup of order ( 2d ).So, for each ( d ) from 1 to 12, we have two subgroups: cyclic and dihedral, except when ( d = 1 ), where the dihedral subgroup would be the same as the cyclic subgroup (trivial). So, actually, for ( d = 1 ), only one subgroup.Therefore, the number of subgroups would be:- 1 trivial subgroup (order 1)- For each ( d = 2, 3, 4, 6, 12 ): one cyclic and one dihedral subgroup.So that's 1 + (5 divisors) * 2 subgroups = 1 + 10 = 11 subgroups? Wait, no, because for each ( d ), we have one cyclic and one dihedral, but for ( d = 12 ), the dihedral subgroup is the entire group, which we already count as a subgroup.Wait, perhaps I need to think differently. Let me recall that the number of subgroups of ( D_n ) is equal to the number of divisors of ( n ) plus the number of divisors of ( 2n ) that are greater than 2. Hmm, not sure.Alternatively, I remember that the number of subgroups of ( D_n ) is equal to the number of divisors of ( n ) plus the number of divisors of ( 2n ) that are greater than 2. Wait, maybe that's not correct.Wait, perhaps it's better to recall that in ( D_n ), the number of cyclic subgroups is equal to the number of divisors of ( n ), and the number of dihedral subgroups is equal to the number of divisors of ( n ) that are greater than 1.Wait, no, that's not precise. Let me think again.In ( D_n ), the cyclic subgroups correspond to the rotations, which form a cyclic group of order ( n ). So, the number of cyclic subgroups is equal to the number of divisors of ( n ). For each divisor ( d ) of ( n ), there is exactly one cyclic subgroup of order ( d ).Additionally, the dihedral subgroups correspond to the symmetries of regular ( d )-gons for each divisor ( d ) of ( n ). So, for each divisor ( d ) of ( n ), there is a dihedral subgroup of order ( 2d ). However, when ( d = 1 ), the dihedral subgroup is trivial, which is the same as the cyclic subgroup of order 1.Therefore, the total number of subgroups is equal to the number of cyclic subgroups plus the number of dihedral subgroups. Since cyclic subgroups are ( tau(n) ) (number of divisors of ( n )), and dihedral subgroups are also ( tau(n) ), but we have to subtract 1 because the trivial subgroup is counted in both.So, total subgroups = ( tau(n) + tau(n) - 1 = 2tau(n) - 1 ).For ( n = 12 ), ( tau(12) = 6 ) (divisors 1, 2, 3, 4, 6, 12). So, total subgroups would be ( 2*6 - 1 = 11 ).Wait, so 11 subgroups in total.Let me list them:Cyclic subgroups:1. Order 1: trivial2. Order 2: generated by a rotation of 180 degrees3. Order 3: generated by a rotation of 120 degrees4. Order 4: generated by a rotation of 90 degrees5. Order 6: generated by a rotation of 60 degrees6. Order 12: the entire rotation subgroupDihedral subgroups:1. Order 2: same as cyclic subgroup of order 2? Wait, no. Wait, dihedral subgroups are generated by a reflection and a rotation. So, for each divisor ( d ), the dihedral subgroup of order ( 2d ) is generated by a reflection and a rotation of order ( d ).But in ( D_{12} ), the dihedral subgroups would be:1. Order 2: generated by a reflection (but wait, reflections have order 2, so each reflection generates a cyclic subgroup of order 2, not a dihedral subgroup)2. Order 4: generated by a reflection and a rotation of order 23. Order 6: generated by a reflection and a rotation of order 34. Order 8: generated by a reflection and a rotation of order 45. Order 12: generated by a reflection and a rotation of order 66. Order 24: the entire group, which is dihedralWait, but in ( D_{12} ), the dihedral subgroups of order 2d where d divides 12. So, for d=1: order 2 (but that's just a reflection, which is cyclic)d=2: order 4d=3: order 6d=4: order 8d=6: order 12d=12: order 24But in ( D_{12} ), the dihedral subgroups of order 2d where d is a divisor of 12. So, for each d, we have a dihedral subgroup of order 2d.But wait, the dihedral subgroup of order 2 is just a reflection, which is cyclic. So, perhaps the dihedral subgroups of order greater than 2.So, for d=2: order 4d=3: order 6d=4: order 8d=6: order 12d=12: order 24So, that's 5 dihedral subgroups of order 4,6,8,12,24.And cyclic subgroups: orders 1,2,3,4,6,12.So, total subgroups:Cyclic: 6Dihedral: 5Total: 11Yes, that makes sense.So, the number of distinct subgroups is 11.Now, describing their structure:Cyclic subgroups:- Order 1: trivial group- Order 2: cyclic group ( C_2 )- Order 3: cyclic group ( C_3 )- Order 4: cyclic group ( C_4 )- Order 6: cyclic group ( C_6 )- Order 12: cyclic group ( C_{12} )Dihedral subgroups:- Order 4: dihedral group ( D_2 ) (which is isomorphic to the Klein four-group ( V_4 ))- Order 6: dihedral group ( D_3 ) (which is isomorphic to the symmetric group ( S_3 ))- Order 8: dihedral group ( D_4 )- Order 12: dihedral group ( D_6 )- Order 24: dihedral group ( D_{12} ) itselfWait, but hold on. For dihedral subgroups, their orders are 4,6,8,12,24. So, their structures are:- Order 4: ( D_2 cong C_2 times C_2 ) (Klein four-group)- Order 6: ( D_3 cong S_3 )- Order 8: ( D_4 )- Order 12: ( D_6 )- Order 24: ( D_{12} )So, that's the structure.Therefore, the number of distinct subgroups is 11, with the cyclic ones being ( C_1, C_2, C_3, C_4, C_6, C_{12} ) and the dihedral ones being ( D_2, D_3, D_4, D_6, D_{12} ).Moving on to the second part: the gender polytope is described by the vertices of a 4-dimensional hypercube projected into three dimensions. The hypercube is represented by coordinates ( {(x_1, x_2, x_3, x_4) ,|, x_i in {0, 1}} ). I need to determine the number of distinct planes formed by the intersection of three of these projected vertices, where each plane represents a different expression of gender. How does this relate to the symmetries of the original group ( G )?First, let's understand the hypercube. A 4-dimensional hypercube, or tesseract, has 16 vertices, each with coordinates ( (x_1, x_2, x_3, x_4) ) where each ( x_i ) is 0 or 1.When projected into 3D, these vertices are mapped to points in 3D space. The projection can be done in various ways, but a common method is to use a perspective projection or an orthographic projection. However, the exact projection method isn't specified here, so I might need to make some assumptions.But regardless of the projection method, each vertex in 4D will correspond to a point in 3D. The key is that the projection is likely to preserve some of the combinatorial structure of the hypercube.Now, the question is about the number of distinct planes formed by the intersection of three projected vertices. Each plane represents a different expression of gender.First, let's think about how many planes can be formed by three points in 3D space. In general, three non-collinear points define a plane. However, in this case, the points are projections of the hypercube vertices, so some sets of three points might lie on the same plane due to the symmetries of the hypercube.But the problem specifies "distinct planes," so we need to count the number of unique planes that can be formed by any three projected vertices.But wait, the hypercube has 16 vertices. The number of possible planes is the number of combinations of three vertices, minus those that are collinear. However, in 3D space, three points are collinear only if they lie on a straight line. So, first, I need to find how many sets of three points are collinear in the projection.But without knowing the exact projection, it's hard to say. However, in a typical projection of a hypercube, the number of collinear triples is limited. For example, in an orthographic projection along one axis, each edge of the hypercube projects to a line segment in 3D, and each face projects to a square or a rectangle.Wait, but in 4D, the hypercube has edges, faces, cells, and the whole hypercube. When projected into 3D, edges can overlap or project to lines, faces project to planar figures, etc.But perhaps the key is that each face of the hypercube is a 3D cube, which when projected, can result in planar figures. However, I'm not sure.Alternatively, maybe the number of planes corresponds to the number of 2-dimensional faces (squares) in the hypercube. Each square face would project to a plane in 3D. But a hypercube has 24 square faces. However, when projected, some of these might coincide or overlap, depending on the projection.Wait, but the question is about planes formed by three projected vertices. So, each plane is determined by three points. However, in the hypercube, each square face has four vertices, so each square face would give rise to multiple planes, but in reality, all four points lie on the same plane.But in the projection, if four points lie on the same plane, then any three of them would define the same plane. So, each square face would correspond to one plane in the projection.But a hypercube has 24 square faces, so does that mean 24 planes? But wait, in the projection, some square faces might project to the same plane, especially if the projection is along a symmetry axis.Wait, but in a typical projection, such as projecting along the (1,1,1,1) direction, the number of distinct planes might be more.Alternatively, perhaps the number of planes is related to the number of 2-dimensional subspaces in the hypercube. But I'm not sure.Alternatively, maybe it's related to the number of ways three vertices can lie on a plane, considering the symmetries.But perhaps another approach is needed.Given that the hypercube has 16 vertices, the total number of planes determined by three vertices is ( binom{16}{3} ), but subtracting those that are collinear.However, in 3D space, three points are collinear only if they lie on a straight line. So, how many triples of vertices are collinear in the projection?In the hypercube, two vertices are connected by an edge if they differ in exactly one coordinate. So, in 4D, each edge is a line segment. When projected into 3D, each edge might project to a line segment, but multiple edges could project to the same line.Similarly, in the hypercube, there are also space diagonals, face diagonals, etc. So, in 4D, each edge is a 1D line, but in projection, multiple edges might overlap or project to the same line.Therefore, the number of collinear triples would be equal to the number of lines in the projection multiplied by the number of ways to choose three points on each line.But without knowing the exact projection, it's difficult to compute. However, perhaps the question is more abstract, considering the projection as a linear transformation, so that the combinatorial structure is preserved in terms of incidences.Alternatively, perhaps the number of planes is related to the number of 2-dimensional faces of the hypercube, which is 24. Each face is a square, which when projected, lies on a plane. So, each square contributes one plane.But in the projection, some squares might project to the same plane. For example, if the projection is along a symmetry axis, multiple squares could project onto the same plane.But in the case of a generic projection, perhaps each square projects to a distinct plane. However, in reality, some squares are parallel or symmetric, so their projections might coincide.Wait, but in 3D, two distinct squares in the hypercube can project to the same plane if they are symmetric with respect to the projection direction.Given that the hypercube has a high degree of symmetry, it's likely that multiple squares project to the same plane.But without a specific projection, it's hard to say. However, perhaps the question is more about the combinatorial structure rather than the geometric projection.Wait, maybe the planes correspond to the 2-dimensional subspaces of the hypercube, which are the square faces. Each square face is a 2-dimensional subspace, and when projected, each corresponds to a plane.But the hypercube has 24 square faces, so does that mean 24 planes? But in the projection, some of these might coincide.Alternatively, perhaps the number of planes is equal to the number of 2-dimensional coordinate planes in 4D, which is 6 (xy, xz, xw, yz, yw, zw). But when projected into 3D, these might correspond to different planes.Wait, but the projection is into 3D, so each 2D coordinate plane in 4D would project to a plane in 3D, but depending on the projection, multiple 2D planes might project to the same plane.Alternatively, perhaps the number of planes is related to the number of ways to choose three coordinates out of four, which is 4, but that doesn't seem right.Wait, another approach: the hypercube has 16 vertices. Each vertex is connected to 4 others via edges. Each edge is part of multiple squares.But in 3D projection, each square can be represented as a planar figure. So, the number of distinct planes would correspond to the number of squares in the hypercube, but considering that some squares project to the same plane.But without knowing the projection, it's hard to count.Wait, maybe the question is more about the combinatorial aspect, not the geometric. It says "the number of distinct planes that can be formed by the intersection of three of these projected vertices." So, it's about how many unique planes can be formed by any three vertices in the projection.In 3D space, the maximum number of planes is ( binom{16}{3} ), but many of these will coincide because of the hypercube's symmetry.But perhaps the number of distinct planes is equal to the number of orbits of the group ( G ) acting on the set of planes. Since ( G ) is the symmetry group of the hypercube, which is actually the hyperoctahedral group, but in this case, the group is ( D_{12} ), which is different.Wait, hold on. The group ( G ) is ( D_{12} ), which is the symmetry group of a 12-gon, but the hypercube's symmetry group is much larger. So, perhaps the projection is done in a way that the symmetries of ( G ) are embedded into the hypercube's symmetries.But I might be overcomplicating.Alternatively, perhaps the number of planes is related to the number of subgroups of ( G ), which is 11. But that might not be directly.Wait, the problem says "how does this relate to the symmetries of the original group ( G )." So, perhaps the number of planes is equal to the number of subgroups or something similar.But I'm not sure. Let me think differently.Each plane is determined by three points. In the hypercube, each square face has four points, so each square contributes ( binom{4}{3} = 4 ) planes, but all these planes are the same because the four points lie on a single plane.Therefore, each square face corresponds to one plane. Since the hypercube has 24 square faces, there are 24 planes. However, in the projection, some of these might project to the same plane in 3D.But how many distinct planes would that be?In a typical projection, such as orthographic projection along one axis, the hypercube projects to a 3D cube with some internal structure. In this case, the number of distinct planes would correspond to the number of square faces in the projection.Wait, in an orthographic projection along, say, the w-axis, the hypercube projects to a 3D cube, with each square face of the hypercube projecting to either a face of the cube or an internal square.But in this case, the number of distinct planes would be equal to the number of square faces in the projection, which includes both the outer faces and the inner faces.A 3D cube has 6 faces, but when considering the projection of the hypercube, each original square face either becomes a face of the cube or an internal square. So, the number of distinct planes would be more than 6.Wait, actually, in the projection, each square face of the hypercube that is orthogonal to the projection direction becomes a square in the projection, while those parallel to the projection direction become lines.But no, actually, in orthographic projection, faces orthogonal to the projection direction are collapsed into lines, while faces parallel to the projection direction are preserved.Wait, no, actually, in orthographic projection along the w-axis, the x, y, z coordinates are preserved, and the w-coordinate is ignored. So, each square face that lies in a constant w-coordinate plane is projected to a square in 3D. Additionally, each square face that varies in w-coordinate is projected to a rectangle or another shape.Wait, perhaps it's better to think that in the projection, each square face of the hypercube either projects to a square or a rectangle or a line, depending on its orientation.But regardless, the number of distinct planes in the projection would correspond to the number of square faces in the hypercube that project to distinct planes in 3D.But without a specific projection, it's hard to say. However, perhaps the number is 24, as each square face corresponds to a plane, but in projection, some coincide.Alternatively, perhaps the number is 16 choose 3 minus the number of collinear triples, but that would be a huge number, which doesn't seem right.Wait, another thought: in the hypercube, each pair of vertices is connected by an edge or not. The number of planes is determined by three non-collinear points. So, if three points are not on the same edge or face diagonal, they form a plane.But in the hypercube, three points can lie on a square face, a cube face, or a tetrahedral configuration.Wait, perhaps the number of planes is equal to the number of 2-dimensional faces (squares) plus the number of 3-dimensional faces (cubes) times the number of planes per cube.But a cube has 6 faces, each a square, so each cube contributes 6 planes. The hypercube has 8 cubic cells, so 8*6=48 planes. But each square is shared by two cubes, so total unique planes would be 24.Wait, that's the number of square faces in the hypercube. So, 24 planes.But in the projection, each square face projects to a plane, but some might coincide.Wait, but if the projection is such that no two square faces project to the same plane, then the number of distinct planes would be 24. However, in reality, due to the projection, some square faces might project to the same plane.For example, in an orthographic projection along a space diagonal, multiple square faces might project to the same plane.But without knowing the exact projection, it's hard to determine.Alternatively, perhaps the number of planes is equal to the number of 2-dimensional subspaces in the hypercube, which is 24, as each square face is a 2-dimensional subspace.But in the projection, each 2-dimensional subspace might correspond to a plane in 3D, but some could overlap.Alternatively, perhaps the number of planes is equal to the number of ways to choose three coordinates out of four, which is 4, but that doesn't make sense because 4 is too small.Wait, another approach: in 3D space, the number of distinct planes determined by n points is at most ( binom{n}{3} ), but with many coinciding due to collinearity and coplanarity.In our case, n=16, but the points are projections of hypercube vertices, which have a lot of structure.I recall that in a hypercube, the number of planes (2-dimensional affine subspaces) can be calculated based on the combinatorial structure.But I'm not sure about the exact number.Wait, perhaps the number of planes is equal to the number of 2-dimensional faces, which is 24, as each square face is a plane.But in the projection, each square face projects to a plane, but some might overlap.However, in a typical projection, such as orthographic, the number of distinct planes would be equal to the number of square faces, as each square face is projected to a distinct plane.But I'm not entirely certain.Alternatively, perhaps the number of planes is equal to the number of 2-dimensional coordinate planes in 4D, which is 6, but that seems too low.Wait, another thought: the hypercube has 16 vertices, each connected to 4 edges. Each edge is part of 3 squares. So, each edge is shared by 3 square faces.But in the projection, each edge projects to a line, and each square projects to a plane.But I'm not sure.Wait, maybe it's better to think about the number of planes in terms of the hypercube's structure.In the hypercube, the number of 2-dimensional faces is 24. Each 2-face is a square. So, each square is a plane in 4D. When projected into 3D, each square projects to a plane in 3D.However, depending on the projection, some squares might project to the same plane.But if the projection is such that no two squares project to the same plane, then the number of distinct planes would be 24.But in reality, due to the projection, some squares will project to the same plane.For example, in an orthographic projection along one axis, the hypercube projects to a 3D cube, and each square face of the hypercube that is orthogonal to the projection axis projects to a square in 3D, while those parallel to the projection axis project to lines.Wait, no, actually, in orthographic projection along the w-axis, the x, y, z coordinates are preserved, so each square face that lies in a plane where w is constant projects to a square in 3D. Additionally, each square face that varies in w projects to a rectangle or another shape.But in this case, the number of distinct planes would be equal to the number of square faces in the hypercube that are orthogonal to the projection direction, which is 6 (since each pair of opposite faces in the hypercube along the w-axis projects to a square in 3D).But that seems too low.Alternatively, perhaps the number of planes is equal to the number of square faces in the hypercube, which is 24, but in the projection, each square face projects to a unique plane, so the number is 24.But I'm not sure.Alternatively, perhaps the number of planes is equal to the number of 3D cubes in the hypercube times the number of planes per cube, but each cube has 6 planes, and there are 8 cubes, but each plane is shared by multiple cubes.Wait, the hypercube has 8 cubic cells, each with 6 square faces, so 8*6=48, but each square face is shared by two cubes, so total unique square faces are 24.So, 24 square faces, each corresponding to a plane.Therefore, the number of distinct planes is 24.But in the projection, each square face projects to a plane, but depending on the projection, some might overlap.However, if the projection is such that no two square faces project to the same plane, then the number is 24.But in reality, due to the projection, some square faces might project to the same plane.For example, in an orthographic projection along a space diagonal, multiple square faces might project to the same plane.But without knowing the exact projection, it's hard to say.However, perhaps the question is more abstract, considering that each plane corresponds to a square face, and thus the number is 24.But the problem says "the number of distinct planes that can be formed by the intersection of three of these projected vertices."So, perhaps it's considering all possible planes, not just those corresponding to square faces.In that case, the number would be larger.Wait, but in the hypercube, any three vertices that are not collinear define a plane. So, the number of planes is equal to the number of combinations of three vertices minus the number of collinear triples.But in 4D, three vertices are collinear only if they lie on a straight line, which in the hypercube corresponds to edges or space diagonals.In the hypercube, each edge is a line segment with two vertices, so no three vertices are collinear along an edge. However, there are space diagonals that pass through four vertices. For example, the main space diagonal from (0,0,0,0) to (1,1,1,1) passes through four vertices: (0,0,0,0), (1,1,1,1), but wait, actually, in the hypercube, each space diagonal connects two vertices, so there are no three collinear vertices along a space diagonal.Wait, actually, in the hypercube, any three vertices are either on a square face, a cube face, or form a tetrahedron.Wait, no, in the hypercube, three vertices can lie on a square face, which is a 2D subspace, or they can lie on a cube face, which is a 3D subspace, or they can form a triangle in 4D space.But in the projection, three vertices might lie on a plane even if they don't lie on a square face in the hypercube.Therefore, the number of planes is more than 24.But calculating this is complex.Alternatively, perhaps the number of planes is equal to the number of 2-dimensional subspaces of the hypercube, which is 24, as each square face is a 2-dimensional subspace.But in the projection, each 2-dimensional subspace corresponds to a plane in 3D.But again, without knowing the projection, it's hard to say.Wait, perhaps the number of planes is equal to the number of ways to choose three coordinates out of four, which is 4, but that seems too low.Alternatively, perhaps the number is 58, which is the number of 2-dimensional faces in the hypercube's projection, but I'm not sure.Wait, actually, in the hypercube, the number of 2-dimensional faces is 24, as each square face is a 2D face.But in the projection, each 2D face projects to a plane, but some might overlap.However, in a typical projection, such as orthographic, the number of distinct planes would be equal to the number of square faces, which is 24.But I'm not entirely certain.Alternatively, perhaps the number of planes is equal to the number of 3D cubes in the hypercube times the number of planes per cube, but each cube has 6 planes, and there are 8 cubes, but each plane is shared by multiple cubes.Wait, but that would be 8*6=48, but each plane is shared by two cubes, so 24 unique planes.So, that's consistent with the number of square faces.Therefore, perhaps the number of distinct planes is 24.But the problem says "the number of distinct planes that can be formed by the intersection of three of these projected vertices."So, each plane is determined by three vertices, but in the hypercube, each square face has four vertices, so any three of them define the same plane.Therefore, the number of distinct planes is equal to the number of square faces, which is 24.But in the projection, some square faces might project to the same plane, so the number could be less.However, if the projection is such that each square face projects to a unique plane, then the number is 24.But the problem doesn't specify the projection method, so perhaps it's assuming a generic projection where each square face projects to a unique plane.Therefore, the number of distinct planes is 24.Now, relating this to the symmetries of the original group ( G ), which is ( D_{12} ).The group ( D_{12} ) has 24 elements, which is the same as the number of planes we just calculated.Therefore, perhaps each plane corresponds to an element of the group, or the number of planes is equal to the order of the group.But more precisely, the number of distinct planes is 24, which is equal to the order of ( G ). So, the number of planes relates to the symmetries of ( G ) in that it matches the group's order.Alternatively, perhaps the symmetries of ( G ) correspond to the symmetries of the hypercube's projection, but I'm not sure.Wait, the hypercube's symmetry group is much larger than ( D_{12} ). The hypercube's symmetry group is the hyperoctahedral group of order ( 2^4 times 4! = 384 ), which is much larger than ( D_{12} ) of order 24.Therefore, perhaps the connection is that the number of planes is equal to the order of ( G ), which is 24.So, putting it all together:1. The number of distinct subgroups of ( G ) is 11, consisting of cyclic groups ( C_1, C_2, C_3, C_4, C_6, C_{12} ) and dihedral groups ( D_2, D_3, D_4, D_6, D_{12} ).2. The number of distinct planes formed by three projected vertices is 24, which is equal to the order of the group ( G ), indicating a relationship through their respective counts.But wait, in the first part, the number of subgroups is 11, and in the second part, the number of planes is 24, which is the order of ( G ). So, the relation is that the number of planes equals the order of ( G ).Therefore, the final answers are:1. 11 subgroups.2. 24 planes, relating to the order of ( G ).But let me double-check the first part.Yes, for ( D_{12} ), the number of subgroups is 11: 6 cyclic and 5 dihedral (excluding the trivial dihedral subgroup which coincides with the cyclic one).And for the second part, the number of planes is 24, matching the order of ( G ).So, I think that's the answer.</think>

🧠:<think>Okay, so I'm trying to solve this problem about a decentralized finance (DeFi) application's smart contract. The problem involves calculating the interest rate based on a given function and then analyzing the function's critical points and concavity. Let me break it down step by step.First, let's tackle the first sub-problem. I need to calculate the interest rate ( r(U) ) given specific values for ( r_0 ), ( alpha ), ( U_{text{min}} ), ( U_{text{max}} ), and ( U ). The formula provided is:[ r(U) = r_0 + alpha left( frac{U - U_{text{min}}}{U_{text{max}} - U_{text{min}}} right)^2 ]The given values are:- ( r_0 = 0.02 ) (which is 2%)- ( alpha = 0.18 ) (which is 18%)- ( U_{text{min}} = 0.1 )- ( U_{text{max}} = 0.9 )- Current ( U = 0.5 )So, plugging these into the formula, let me compute each part step by step.First, compute the numerator of the fraction inside the square: ( U - U_{text{min}} ). That would be ( 0.5 - 0.1 = 0.4 ).Next, compute the denominator: ( U_{text{max}} - U_{text{min}} ). That's ( 0.9 - 0.1 = 0.8 ).So, the fraction is ( frac{0.4}{0.8} ). Simplifying that, ( 0.4 ÷ 0.8 = 0.5 ).Now, square this value: ( (0.5)^2 = 0.25 ).Multiply this squared value by ( alpha ): ( 0.18 × 0.25 ). Let me calculate that. 0.18 times 0.25 is 0.045.Finally, add this result to the base interest rate ( r_0 ): ( 0.02 + 0.045 = 0.065 ).So, converting that back to a percentage, 0.065 is 6.5%. Therefore, the interest rate ( r(U) ) when ( U = 0.5 ) is 6.5%.Wait, let me double-check my calculations to make sure I didn't make any mistakes.- ( U - U_{text{min}} = 0.5 - 0.1 = 0.4 ) ✔️- ( U_{text{max}} - U_{text{min}} = 0.9 - 0.1 = 0.8 ) ✔️- ( frac{0.4}{0.8} = 0.5 ) ✔️- ( (0.5)^2 = 0.25 ) ✔️- ( 0.18 × 0.25 = 0.045 ) ✔️- ( 0.02 + 0.045 = 0.065 ) which is 6.5% ✔️Okay, that seems correct.Now, moving on to the second sub-problem. I need to determine the critical points and analyze the concavity of the interest rate function ( r(U) ). Specifically, I have to find the value of ( U ) that maximizes the interest rate within the interval ([U_{text{min}}, U_{text{max}}]).First, let me write down the function again:[ r(U) = r_0 + alpha left( frac{U - U_{text{min}}}{U_{text{max}} - U_{text{min}}} right)^2 ]This is a quadratic function in terms of ( U ), but scaled and shifted. Since it's a quadratic function, its graph is a parabola. The coefficient of the quadratic term will determine whether it opens upwards or downwards, which in turn affects the concavity and the presence of a maximum or minimum.Looking at the function, the term ( left( frac{U - U_{text{min}}}{U_{text{max}} - U_{text{min}}} right)^2 ) is squared, so it's always non-negative. The coefficient ( alpha ) is positive (0.18), so the entire quadratic term is non-negative and adds to the base rate ( r_0 ). Therefore, the function ( r(U) ) is a parabola opening upwards, meaning it has a minimum point and is concave upwards.Wait, hold on. If the coefficient is positive, the parabola opens upwards, so it has a minimum, not a maximum. That suggests that the interest rate function doesn't have a maximum within the interval unless it's at the endpoints. Hmm, but the question says to find the value of ( U ) that maximizes the interest rate within the interval. So, maybe I need to think again.Wait, perhaps I made a mistake in interpreting the function. Let me analyze the function more carefully.The function is:[ r(U) = r_0 + alpha left( frac{U - U_{text{min}}}{U_{text{max}} - U_{text{min}}} right)^2 ]Let me denote ( x = frac{U - U_{text{min}}}{U_{text{max}} - U_{text{min}}} ). Then, the function becomes:[ r(U) = r_0 + alpha x^2 ]So, ( x ) ranges from 0 to 1 because ( U ) ranges from ( U_{text{min}} ) to ( U_{text{max}} ). Therefore, ( x ) is in [0,1].So, the function ( r(U) ) is a quadratic function in ( x ), which is a parabola opening upwards because ( alpha ) is positive. Therefore, the minimum value occurs at the vertex, and the maximum occurs at the endpoints of the interval.Wait, so if it's a parabola opening upwards, the minimum is at the vertex, and the maximum would be at the endpoints, either at ( x = 0 ) or ( x = 1 ). Let's compute the interest rate at both endpoints.At ( x = 0 ), which corresponds to ( U = U_{text{min}} = 0.1 ):[ r(U) = r_0 + alpha (0)^2 = r_0 = 0.02 ]At ( x = 1 ), which corresponds to ( U = U_{text{max}} = 0.9 ):[ r(U) = r_0 + alpha (1)^2 = 0.02 + 0.18 = 0.20 ]So, the interest rate is 20% at ( U = 0.9 ), which is higher than at ( U = 0.1 ) (2%). Therefore, the maximum interest rate occurs at ( U = U_{text{max}} = 0.9 ).But wait, the question asks for the value of ( U ) that maximizes the interest rate within the interval. So, according to this, it's at ( U = 0.9 ).However, let me think again about the function. Since it's a quadratic function, the vertex is the minimum point. The vertex occurs where the derivative is zero. Let's compute the derivative to find the critical points.Taking the derivative of ( r(U) ) with respect to ( U ):First, express ( r(U) ) as:[ r(U) = r_0 + alpha left( frac{U - U_{text{min}}}{U_{text{max}} - U_{text{min}}} right)^2 ]Let me denote ( a = U_{text{min}} ) and ( b = U_{text{max}} ) for simplicity. Then,[ r(U) = r_0 + alpha left( frac{U - a}{b - a} right)^2 ]Taking the derivative with respect to ( U ):[ frac{dr}{dU} = 2 alpha left( frac{U - a}{b - a} right) cdot frac{1}{b - a} ]Simplify:[ frac{dr}{dU} = frac{2 alpha}{(b - a)^2} (U - a) ]Set the derivative equal to zero to find critical points:[ frac{2 alpha}{(b - a)^2} (U - a) = 0 ]Since ( alpha ) and ( (b - a)^2 ) are positive, the only solution is when ( U - a = 0 ), i.e., ( U = a = U_{text{min}} ).So, the critical point is at ( U = U_{text{min}} ), which is a minimum because the parabola opens upwards.Therefore, within the interval ([U_{text{min}}, U_{text{max}}]), the function ( r(U) ) has a minimum at ( U = U_{text{min}} ) and increases as ( U ) moves away from ( U_{text{min}} ) towards ( U_{text{max}} ). Hence, the maximum interest rate occurs at ( U = U_{text{max}} ).To confirm, let's compute the second derivative to check concavity.The second derivative of ( r(U) ) with respect to ( U ):[ frac{d^2r}{dU^2} = frac{2 alpha}{(b - a)^2} ]Since ( alpha ) and ( (b - a)^2 ) are positive, the second derivative is positive, meaning the function is concave upwards everywhere. Therefore, the function has a minimum at ( U = U_{text{min}} ) and is increasing on the interval ([U_{text{min}}, U_{text{max}}]). Thus, the maximum occurs at ( U = U_{text{max}} ).Wait, but the problem says "find the value of ( U ) that maximizes the interest rate within the interval ([U_{text{min}}, U_{text{max}}])". So, according to this analysis, it's at ( U = U_{text{max}} = 0.9 ).But let me think again. The function is quadratic, so it's symmetric around its vertex. However, in this case, the vertex is at ( U = U_{text{min}} ), and since the parabola opens upwards, the function increases as ( U ) increases beyond ( U_{text{min}} ). Therefore, the maximum in the interval would indeed be at ( U = U_{text{max}} ).Alternatively, if the function were a downward-opening parabola, the maximum would be at the vertex. But since it's upward-opening, the maximum is at the endpoint.Therefore, the critical point is at ( U = U_{text{min}} ), which is a minimum, and the maximum is at ( U = U_{text{max}} ).So, to summarize the second sub-problem:- The function ( r(U) ) has a critical point at ( U = U_{text{min}} ), which is a minimum.- The function is concave upwards throughout the interval.- The maximum interest rate occurs at ( U = U_{text{max}} = 0.9 ).Wait, but let me make sure I didn't misinterpret the function. The function is ( r(U) = r_0 + alpha left( frac{U - U_{text{min}}}{U_{text{max}} - U_{text{min}}} right)^2 ). So, as ( U ) increases from ( U_{text{min}} ) to ( U_{text{max}} ), the term ( frac{U - U_{text{min}}}{U_{text{max}} - U_{text{min}}} ) increases from 0 to 1, and its square increases from 0 to 1. Therefore, the interest rate increases from ( r_0 ) to ( r_0 + alpha ). So, yes, the maximum is at ( U = U_{text{max}} ).Therefore, the value of ( U ) that maximizes the interest rate is ( U = 0.9 ).Wait, but let me think about the concavity again. The second derivative is positive, so the function is concave upwards, meaning it curves upward, like a U-shape. So, the function has a minimum at ( U = U_{text{min}} ) and increases on either side, but since we're only considering the interval from ( U_{text{min}} ) to ( U_{text{max}} ), the function is increasing throughout this interval. Therefore, the maximum is indeed at ( U = U_{text{max}} ).So, to recap:1. Calculated ( r(U) ) at ( U = 0.5 ) and got 6.5%.2. Analyzed the function and found that the critical point is at ( U = U_{text{min}} ), which is a minimum, and the function is concave upwards. Therefore, the maximum interest rate occurs at ( U = U_{text{max}} = 0.9 ).I think that's correct. Let me just check if I considered all aspects.Wait, another thought: sometimes, in optimization problems, especially in economics, the maximum might not always be at the endpoints, but in this case, since the function is strictly increasing in the interval, the maximum is indeed at the upper endpoint.Yes, because the derivative is positive for all ( U > U_{text{min}} ), meaning the function is increasing throughout the interval. Therefore, the maximum is at ( U = U_{text{max}} ).Okay, I think I've covered all the necessary steps and checked my work. I'm confident in my answers now.</think>