Can you help with question 5? Equation (3) I need to verify is attached. CHAPTER 3 / STATIC OPTIMIZATION4. Find the functions x (r) and y"(r) such that x =x*(r) and y = y (r) solve the problemmax(-x – xy – 2y²+2rx + 2ry)where r is a parameter. Verify equation (3).SM 5. Find the solutions x*(r, s) and y*(r, s) of the problemmax f (x, y, r, s) =max(r'x + 3s*y - x - 8y²)where r and s are parameters. Verify equation (3).SM 6. (a) Suppose the production function in Example 2 isF(v,..., Vn) = a¡ In(v1 + 1)+ · + a, In(vn + 1)where a1, ..., an are positive constants and p > qi/a¡ for i = 1,., n. Find the profitmaximizing choice of input quantities.(b) Verify the envelope result (3) w.r.t. p, each q1 and each a,.3.2 Local Extreme PointsSuppose one is trying to find the maximum of a function that is not concave, or a minimumof a function that is not convex. Then Theorem 3.1.2 cannot be used. Instead, one possibleprocedure is to identify local extreme points, and then compare the values of the functionat different local extreme points in the hope of finding a global maximum (or minimum).The point x is a local maximum point of f in S if ƒ(x) < ƒ(x*)forall x in S sufficientlyclose to x*. More precisely, the requirement is that there exists a positive number r such thatf (x) < f(x*) for all x in S with ||x- x' || Later we shall givé more efficient methods for finding extremepoints of such functions.Envelope Theorem for Unconstrained MaximaThe objective function in economic optimization problems usually involves parameters likeprices in addition to choice variables like quantities. Consider an objective function with aparameter vector r of the form f(x, r) = f(x1,.. , Xn, ri,..., rk), where x ESCR" andre R. For each fixed r suppose we have found the maximum of f(x, r) when x varies inS. The maximum value of f (x, r) usually depends on r. We denote this value by f*(r) andcall f the value function. Thus,%3D(1),fXES= max f(x, r)(the value function)()The vector x that maximizes f (x, r) depends on r and is denoted by x* (r).' Thenf*(r) =f(x*(r), r).How does f*(r) vary as the jth parameter r; changes? Provided that f* is differentiablewe have the following so-called envelope result:(1), feraf (x, r)j = 1, ., k(3)%3D(1),X=1rNote that on the right-hand side we differentiate f w.r.t. its (n + j)th argument, which isri, and evaluate the derivative at (x*(r), r).When the parameter r; changes, f*(r) changes for two reasons. First, a change inr; changes x*(r). Second, f(x*(r), r) changes directly because the variable r; changes.1 There may be several choices of x that maximize f (x, r) for a given parameter vector r. Then welet x* (r) denote one of these choices, and try to select x for different values of r so that x*(r) is adifferentiable function of r.10:30 PM4/3/2021

Name: Can you help with question 5? Equation (3) I need to verify is attache
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Description: Can you help with question 5? Equation (3) I need to verify is attached. CHAPTER 3 / STATIC OPTIMIZATION4. Find the functions x (r) and y"(r) such that x =x*(r) and y = y (r) solve the problemmax(-x – xy – 2y²+2rx + 2ry)where r is a parameter. Verify