A manufacturer finds that the revenue generated by selling x

A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function R(x) = 60x - 0.4x^2, where the revenue R(x) is measured in dollars. What is the maximum revenue, and how many units should be to obtain this maximum?

Solution

The revenue function for the manufacturer is R(x) = 60x -0.4x². Then dR/dx = 60-0.8x and d²R/dx² = 0.8. We know that, if R(x) is to be maximum, then dR/dx = 0 and d²R/dx² should be negative. Here, if dR/dx = 0, then 60-0.8x = 0 or, 0.8x = 60 so that x = 60/0.8 = 75. Also, regardless of the value of x, d²R/dx² is negative. Hence, 75 units should be manufactured to maximize the revenue.