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 $p \le 32$), then the number of books sold per month is $128-4p$. What price should the store charge to maximize its revenue?
To solve this problem, I need to find a function that represents the revenue of the bookstore, and then find the value of $p$ that maximizes that function. The revenue is the product of the price and the number of books sold, so the revenue function is $R(p) = p(128-4p)$. To maximize $R(p)$, I need to find the value of $p$ that makes $R'(p) = 0$, or where $R(p)$ has a critical point. To find $R'(p)$, I can use the product rule: $R'(p) = 128-4p + p(-4) = 128-8p$. Setting this equal to zero and solving for $p$, I get $p = 16$. This means that the revenue is maximized when the price is $16$ dollars. To check if this is indeed the maximum, I can use the second derivative test: $R''(p) = -8$, which is negative for all values of $p$, so $R(p)$ is concave down and $p = 16$ is a maximum point. Therefore, the bookstore should charge $16$ dollars for the book to maximize its revenue. 