Question

The demand function for a certain commodity is

$p\left(x\right)=10-0.001x$,

where p is measured in dollars and x is the number of units produced and sold. The total cost of producing x items is

$C\left(x\right)=50+5x$

Determine the level of production that maximizes the profit.

Answer 1

The profit function is given by

$P\left(x\right)=xp\left(x\right)-C\left(x\right)$

$=x(10-0.001x)-(50+5x)$

$=10x-0.001x^2-50-5x$

$=5x-0.001x^2-50$

Take the derivative of $P\left(x\right)$ :

$P^{\prime }\left(x\right)=5-0.002x$

So, the critical point is

$x=2500$

Since, the second derivative of P(x) is negative, $x=2500$ is a point of maximum.

Hence, the company has the largest profit when $x=2500$.