Question

A manufacturer of tennis rackets finds that the total cost C(x) (in dollars) of manufacturing x rackets/day is given by

C(x) = 900 + 3x + 0.0003x².

Each racket can be sold at a price of p dollars, where p is related to x by the demand equation

p = 5 − 0.0002x.

If all rackets that are manufactured can be sold, find the daily level of production that will yield a maximum profit for the manufacturer. Hint: The revenue is

R(x) = px,

and the profit is

P(x) = R(x) − C(x).

How Many Rackets?

[Hint:10,000 rackets is incorrect]

Answer 1

Step 1)

we know that revenue is R(x) = px

we have p = 5 - 0.0002x

Hence,

we know that profit is P(x) = R(x) - C(x)

we have,

Hence,

Step 2)

we will find the critical points of profit function

we have,

Hence,

equate it to 0 we can write,

Hence we can say that x = 2000 is the critical point of the profit function

Step 3)

we will check x = 2000 is maximum or minimum point

we have,

Hence,

Hence,

As P''(2000) = -0.001 < 0 according to second derivative test we can say that x = 2000 is a maximum point

Hence we can say that profit is maximum at x = 2000

we can write for maximum profit daily level of production is 2000 racket