Problem 1 The purpose of this problem is to provide a numerical example of the following phe- nomenon: If an additional unit of a good is more expensive to produce than the cost of production on average, then the relatively high cost of this marginal unit will pull up the average cost. Similarly, if the marginal unit is cheaper to produce than the cost of production on average, then the relatively low cost of this marginal unit will bring down the average cost. Consider a firm whose total costs of production as a function of its output x are given as TC(x) = 3x2 + 7x + 12 for x > 0. (a) Compute the average-cost function AC(x) and the marginal-cost function MC(x). (b) Compute the derivative AC' (x) of the average-cost function. (c) Show that the average-cost function is minimized at x = 2. Make sure to check the second-order condition. (d) Show the marginal cost exceeds the average cost: MC(x) > AC(x) whenever x > 2. (e) Show that the average-cost function is rising: AC' (x) > 0 whenever x > 2. (f) Show the marginal cost is less than the average cost: MC(x)