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.

The solution given in the book has some error. The solution is created according to the question given in the book.