Question

A profit -maximizing firm has the total-cost function c= x^2 + 2x and sells into a competitive market on which the price is $10. what output should it produce?

hint: Find the derivative and check for local maxima and minima

Answer 1

Answer:

The output produce is 4.

Step-by-step explanation:

Given : A profit -maximizing firm has the total-cost function $c= x^2+2x$ and sells into a competitive market on which the price is $10.

To find : What output should it produce?  

Solution :

The total-cost function $C(x)= x^2+2x$

The revenue function is price into number of item,

So, The revenue function is $R(x)=10x$

The profit function is given by,

$P(x)=R(x)-C(x)$

$P(x)=10x-(x^2+2x)$

$P(x)=10x-x^2-2x$

$P(x)=8x-x^2$

The derivative of the profit function,

$P'(x)=8-2x$

Equate it to zero to get output,

$8-2x=0$

$2x=8$  

$x=4$

For maxima/minima we find the second derivative,

$P''(x)=-2$

As $c''(x)<0$ it is a local maxima.

Therefore, The output produce is 4.