Suppose q > 0 units of a commodity are produced at a total cost of C(q) dollars and an average cost of

In this section, we showed that q = q_c satisfies A'(q_c) = 0 if and only if C'(q_c) = A(q_c); that is, when marginal cost equals average cost. The purpose of this problem is to show that A(q) is minimized when q = q_c.

a. Generally speaking, the cost of producing a commodity increases at an increasing rate as more and more goods are produced. Using this economic principle, what can be said about the sign of C"(q) as q increases?

b. Show that A"(qc) > 0 if and only if C"(q_c) > 0. Then use part (a) to argue that average cost A(q) is minimized when q = q_c.

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A(q) C(q) 9