Cracking The Ap Calculus ab Exam 2018

Thus, in order to find the maximum profit, we need to plug all four values of x into the profit
equation to find which gives the greatest value. We get

P(0)    =   −1,000

P(12)   =   (12)^3  −   48(12)^2    +   720(12) −   1,000   =   2,456

P(20)   =   (20)^3  −   48(20)^2    +   720(20) −   1,000   =   2,200

P(40)   =   (40)^3  −   48(40)^2    +   720(40) −   1,000   =   15,000

Thus, even though we have a relative maximum at x = 12, the absolute maximum profit occurs
when x = 40. The profit at that value is 15,000, or $15 billion dollars.

If we look at a graph of the profit function, we can see that the critical points gave us relative
maximum and minimum, but not the absolute maximum or minimum. This is why it is always
important to check the endpoints of a function in any max/min problem.