Laura owns and operates Aunt Lindas Pecan Pies She has learn

Laura owns and operates Aunt Linda\'s Pecan Pies. She has learned that her profits, P(x), from the sale of x cases of pies, are given by P(x) = 150x - x^2. a. The company will \"break-even\" when the profit is zero. How many cases of pies should Laura sell in order to break-even? (Solve for x when P(x) = 0) b. How many cases of pies should she sell in order to maximize profit? c. What is the maximum profit?

Solution

a) Solution: set Px) =0 and solve for x

=>150x-x^2=0

=> x(150-x) = 0

=> x = 0 or x = 150

150 pie

b) Solution: P\'(x) = 0

i.e. take derivatives for P(x) and set it equal to 0

=> 150 -2x = 0

=> x= 75

75 pies

c) Solution: Substitute x=75 into P(x)

=> P(75) = 150*75 - (75)^2

= 11250-5625

   =5625

Therefore, maximum profit is $5,625