A manufacturer of tennis rackets finds that the total cost C

A manufacturer of tennis rackets finds that the total cost C(x) (in dollars) of manufacturing x rackets/day is given by C(x) = 2000 + 200x + 0.05x^2. Each racket can be sold at a price of p dollars, where p = 500 - 0.2x. If all rackets that are manufactured can be sold, find the daily level of production that maximize profits. 1,000 600 300 200 1,500

Solution

Answer :

The cost function C(x) is given by

C(x) = 2000 + 200x + 0.05x²

The revenue function is R(x) = xp(x) = x( 500 - 0.2x) = 500x - 0.2x²,

and the profit function is P(x) = R(x) - C(x) = 500x - 0.2x² - ( 2000 + 200x + 0.05x² )

                                    P(x) = - 2000 + 300x - 0.25x²

Now we find the derivative P\'(x) = 300 - 0.5x and we set P\'(x)=0 gives x=600

Now we observe that P\'\' (x) = - 0.5 < 0 for all x

so that P(x) is concave down on ( 0, infinty ). This says that the production level x=600 will yield a maximum profit for the manufacturer.