Question

Find the maximum profit that a company can make, if the profit function is given by
$p\left(x\right)=41-72x-18x^2$

Answer 1

The profit function is given by

$p\left(x\right)=41-72x-18x^2$

Differentiating w.r.t $x$, we get

$p^{\prime }\left(x\right)=-72-36x$

Putting $p^{\prime }\left(x\right)=0$

$-72-36x=0$

$\Rightarrow 36x=-72$

$\Rightarrow x=-2$

$\therefore x=-2$ is a critical point.

$\because p^{\prime }\left(x\right)=-72-36x$

Differentiating w.r.t $x$

$p"\left(x\right)=-36$

Since $P"\left(x\right)<0$

$x=-2$ is a point of maxima,

$\because p\left(x\right)=41-72x-18x^2$

$\therefore$ Maximum profit

$=p(-2)$

$=41-72(-2)-18(-2{)}^2$

$=41+144-18\left(4\right)$

$=41+144-72$

$=113$

Hence, the maximum profit is $113$.