A small manufacturer faces a piecewise linear demand curve. For 0≤q≤1,000 units, the price per unit is p(q)=80−0.04q. For q>1,000 units, the price is p(q)=40−0.01(q−1,000). The total cost of producing q units is C(q)=20q+5,000 dollars for all q≥0.
(a) Write an explicit expression for the profit function P(q) on each interval 0≤q≤1,000 and q>1,000. Clearly indicate the domain of each expression.
(b) Find every critical point of P in each interval where the profit function is differentiable.
(c) Evaluate the profit at all critical points and at the endpoint q=1,000 to determine the production level that yields the highest profit over all feasible q.
(d) State the optimal quantity, the price that will be charged, and the maximum profit. Explain how the piecewise nature of the demand curve affects the analysis compared with a single linear demand curve.