Before we even start, notice that there is boundaries to the domain of this story. There are only 200 seats to rent! We will return to that at the end of the problem.
So, we have not answered the question. Sometimes, when a function has a restricted domain, the maximums is NOT where the derivative equals 0, but at an endpoint (where the derivative does not exist). Go back and determine how much Ms. Platinum will need to charge to sell 200 tickets.
Let x = number of times she decreases the price. Write a Revenue function, R(x).
Revenue = (seats sold)(price per seat)
R(x) = (100 + 10x)(40 − 1x)
Maximize the revenue.
R'(x) = 10(40 − x) − 1(100 + 10x) Product Rule
= 300 − 20x
0 = 300 − 20x
x = 15
What does this mean?
She should charge 40 −15 dollars, or $25 and she will sell 100 + 10(15) tickets = 450 tickets
But this is impossible because the theater only has 200 seats.