Question: The production function f(x) gives the number of units of an item that a manufacturing company can produce from x units of raw material. The company buys the raw material at price w dollars per unit and sells all it produces at a price of p dollars per unit. The quantity of raw material that maximizes profit is denoted by x∗.
(a) Do you expect the derivative f'(x) to be positive or negative? Justify your answer.
(b) Explain why the formula π(x) = pf(x) - wx gives the profit π(x) that the company earns as a function of the quantity x of raw materials that it uses.
(c) Evaluate f'(x∗).
(d) Assuming it is nonzero, is f"(x∗) positive or negative?
(e) If the supplier of the raw materials is likely to change the price w, then it is appropriate to treat x∗ as a function of w. Find a formula for the derivative dx∗/dw and decide whether it is positive or negative.
(f) If the price w goes up, should the manufacturing company buy more or less of the raw material?