Question

Find the maximum profit that a company can make, if the profit function is given by

p(x) = 41 − 72x − 18x²

Answer 1

The profit function is given as p(x) = 41 − 72x − 18x².
∴ $p\text{'}\left(x\right)=-72-36x$

$\Rightarrow x=-\frac{72}{36}=-2$
Also,
$p\text{'}\text{'}(-2)=-36<0$
By second derivative test, $x=-2$ is the point of local maxima of p.

∴ $\mathrm{Maximum}\mathrm{profit}=p\left(-2\right)$
$=41-72(-2)-18(-2{)}^2=41+144-72=113$

Hence, the maximum profit that the company can make is 113 units.