optimization

Question

A certain company produces a certain item. The profit, in thousands of dollars, generated by selling x units of this item, is given by P(x)=-.001x^3+3x, where x is measured in thousands.

 

a. Find the maximum profit.

b. How many units must be produced to maximize the profit?

c. After producing how many units will the company start losing money?

Answer 1

To find the maximum take the first derivative respect to x of the P(x) function P'(x) and make it equal to Zero,

P'(x) = -0.003 x^2 + 3

then x = 32 units will be the maximum number to make the maximum profit.

The maximum profit will be obtained replacing x by 32 in the P(x) function, And we obtain: max P(x) = 63.24 and to get the point where the company will begin loosing money, I made the P(x) function equal to Zero and we have the following answers: x=0 ; 55 units.

CHECK!!!!!!!!!!!!!!!!!!!!!!!!!