A manufacturer’s profit in thousands of dollars is modeled by the differentiable function P(x)=15x3−180x2+400x−125, where x is the number of hundreds of units produced.
Use calculus to analyze the firm’s production decisions.
(a) Compute P′(x). Include appropriate units in your answer.
(b) Determine all intervals of x for which the profit is increasing and all intervals for which it is decreasing. Show all work leading to critical points and sign charts or equivalent reasoning.
(c) Interpret your results from part (b) in practical terms for the manufacturer. Specifically, describe the range of production levels that increase profit and those that reduce it.
(d) The firm must choose a single production level at which to operate next quarter. Using your analysis, identify the production quantity that yields a local maximum profit and justify why it is optimal relative to nearby production levels.