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

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

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

Putting this equal to zero we get $x=-2$

Now let's look at second derivative of the given function.

$f^{{}^{\prime \prime }}\left(x\right)=-36$ and as we can see, this is negative for all $x$; hence $f\left(x\right)$ will have it's maxima at $x=-2$.

Maximum profit which the company is going to make is

$p(-2)=41-72\times (-2)-18\times \left({-2}^2\right)$
$=113$.