MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A toy manufacturer sells a novelty item. Market research indicates that when the selling price is p dollars, the monthly demand is q(p)=5000−200p units, where 0≤p≤20.
Revenue R(p) is the product of price and quantity sold.
(a) Write an explicit formula for R(p) on the interval [0,20], and state its domain.
(b) Determine the price that maximizes monthly revenue. Support your answer with appropriate calculus work.
(c) Justify that the price found in part (b) yields the absolute maximum revenue on [0,20] by examining either first–derivative sign changes or concavity.
(d) Compute that maximum monthly revenue, including units in your answer.