How Do You Maximize Profit?

As long as you have the profit function, you can find the maximum using the first derivative of the profit function.

Problem 1 The total profit P(x) (in thousands of dollars) from the sale of x units of a drug is

$latex \displaystyle P(x)=ln \left( -{{x}^{3}}+3{{x}^{2}}+72x+1 \right)$

for x in [0, 10].

a. How many units should be sold to maximize profit?

b. What is the maximum profit?

If you are given the revenue and cost functions, you’ll need to combine those to get the profit function first. Then you can use the first derivative to find the maximum.