A bookstore is deciding what price it should charge for a certain book. After research, the store finds that if the book's price is $p$ dollars (where $p \le 40$), then the number of books sold per month is $120-3p$. What price should the store charge to maximize its revenue?
I need to find the value of $p$ that maximizes the revenue, which is the product of the price and the quantity sold. I can write the revenue as a function of $p$ as follows: $R(p) = p(120-3p)$. To find the maximum of this function, I can use calculus and find the critical points where the derivative is zero or undefined. The derivative of $R(p)$ is $R'(p) = 120-6p$. This is a linear function, so it is never undefined. It is zero when $p = 20$. This means that $p = 20$ is a critical point of $R(p)$. To check if this is a maximum, a minimum, or neither, I can use the second derivative test. The second derivative of $R(p)$ is $R''(p) = -6$. This is negative for all values of $p$, so it means that $R(p)$ is concave down and has a maximum at $p = 20$. Therefore, the bookstore should charge $20$ dollars for the book to maximize its revenue.

