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Due date: 10/26/2023 (11:59 PM) Problem 6 Problem 1. An ice-cream lover has a total of $10 to spend one evening. The price of ice-cream is $p per pint. T person's preferences for buying q pints of ice-cream, leaving a nonnegative amount $(10- pq) to spend Find possible maximum/minimum points for the following functions by non-calculus arguments. other items, are represented by the utility function (a) f(x) =4-2(x+3)2 U(q) = Vq+2\\10-pq, q E [0, 10/p] (a) Find the first-order condition for a utility maximizing quantity of ice-cream, q*. Problem 2 (b) Solve the first-order condition derived in (a) in order to express q* as a function of p. Let f(x) = 200x / be the output quantity of a firm when the input is x. Suppose each unit of output is sold (c) What guarantees that your answer to (a) is really a maximum? at a price of 30 euros. The cost function is C(x) = 15x4/3. Find the profit function (x) and find the value of x 2 0 that maximizes profit. Verify that (x) is concave for x 2 0 and sketch the graph of It(x). Problem 7 Problem 3 (a) Let f(x) = (x2 -2x)et. Find f'(x) and f"(x). (b) Find the zeros of f (where f(x) is 0), as well as the local extreme and inflection points. Sketch the A firm's profit is n(L) = 612 -0.213 when it employs L workers. graph. (a) Find the number of workers that maximizes Q(L) = (L)/L, its average profit per worker. (b) Show that at the optimal value of L in part (a), the marginal product of labour '(L) is equal to the Problem 8 average profit. 4x Problem 4 The function f is defined for all x by f(x) = 2- x 2 + 3 ' The price per unit received by a firm that sells x 2 0 units of output is p = 144 -x, while the cost of (a) Find f'(x) and f"(x). producing x units is C(x) = 3x3 -6x2 + 160x. (b) Determine limx-> too f (x). (a) Show that the marginal cost C'(x) is always positive. (c) Find the possible extreme points and inflection points of f. (b) Show that the profit function is (x) = -3x3+ 5x2 - 16x. (d) Sketch the graph of f. (c) Find the value of x that maximizes profits. Problem 5 Explain why the following functions all have maximum and minimum over the given intervals, and find the maximum and minimum values (a) f(x) = 3x3-4x+1, xe [-1, 3] (b) 8(x) = xe *, xE [-1,5] 2.F