A technology firm models its daily profit (in thousands of dollars) from producing…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A technology firm models its daily profit P (in thousands of dollars) from producing and selling q electronic tablets by
P(q)=−0.002q3+0.3q2+24q−120,
where 0≤q≤500 represents the feasible production levels set by factory capacity.
All quantities are in thousands of dollars or tablets, as indicated.
(a) Compute P(0) and P(500) and interpret these values in the business context.
(b) Determine all production levels q in the closed interval [0,500] at which the profit function has a critical point. Show analytical work leading to your conclusion.
(c) Find the production level that yields the company’s maximum daily profit within the feasible interval, and state the corresponding maximum profit. Justify that your answer gives the absolute maximum on [0,500].
(d) If management can increase factory capacity beyond 500 tablets per day, explain qualitatively how you would investigate whether producing more than 500 tablets could further increase profit.