Example 4: A manufacturer can produce printer paper at a cost of \( \$ 2 \) per ream. The paper has been selling for \( \$ 5 \) per ream, and at that price, consumers have been buying 4,000 reams a month. The manufacturer is planning to raise the price of the paper and estimates that for each \( \$ 1 \) increase in the price, 400 fewer reams will be sold each month.
a) Express the manufacturer's monthly profit as a function of the price at which the reams are sold.
b) What price corresponds to maximum profit? What is the maximum profit?