Find possible maximum/minimum points for the following functions by non-calculus arguments. (a) f(x)=4-2(x+3)2 Problem 2 Let f(x)=200x1/3 be the output quantity of a firm when the input is x. Suppose each unit of output is sold 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 0 that maximizes profit. Verify that (x) is concave for x > 0 and sketch the graph of (x). Problem 3 A firm's profit is (L) = 6L -0.2L when it employs L workers. (a) Find the number of workers that maximizes (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 average profit. Problem 4 The price per unit received by a firm that sells x 0 units of output is p = 144-x, while the cost of producing x units is C(x) = x-6x+160x. (a) Show that the marginal cost C'(x) is always positive. (b) Show that the profit function is 7(x) = -x+5x - 16x. (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) = x-4x+1, x = [1,3] (b) g(x)=xe,x [1,5] An ice-cream lover has a total of $10 to spend one evening. The price of ice-cream is $p per pint. Th person's preferences for buying a pints of ice-cream, leaving a nonnegative amount $(10-pq) to spend o other items, are represented by the utility function U(a)=9+210-pq, q [0,10/p] (a) Find the first-order condition for a utility maximizing quantity of ice-cream, q*. (b) Solve the first-order condition derived in (a) in order to express q* as a function of p. (c) What guarantees that your answer to (a) is really a maximum? Problem 7 (a) Let f(x)=(x-2x)e. 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 graph. Problem 8 4x The function f is defined for all x by f(x) = 2-- x+3 (a) Find f'(x) and f"(x). (b) Determine limo f(x). (c) Find the possible extreme points and inflection points of f. (d) Sketch the graph of f.