Maximum Profit? P = 25-5x

Question

Amy, the owner of Amy's Pottery, can produce china pitchers at a cost of $3 each. She estimates the price function to be p=25-5x, where p is the price at which exactly x pitchers will be sold per week. Find the number of pitchers that she should produce and the price that she should charge in order to maximize profit. Also, find the maximum profit.

Answer 1

Profit = P = R-C = xp -3x = 25x -5x^2 -3x = 22x -5x^2

P' = 22 -10x = 0

x = 22/10 = 2.2 pitchers per week maximizes profit

max P = 22(2.2) -5(2.2)^2 = $24.2 per week