Maximum profit with a cubic function

Statement

A firm’s profit is $\pi(q)=-\dfrac{1}{3}q^{3}-5q^{2}+2000q-350$ (in €). Find the (positive) quantity that maximizes profit and the maximum profit.

Solution

$\pi'(q)=-q^{2}-10q+2000=0\Rightarrow q^{2}+10q-2000=0\Rightarrow (q-40)(q+50)=0$. Discard $q=-50$; with $\pi''=-2q-10$ we get $\pi''(40)=-90<0$: maximum at $q=40$. $\pi(40)=-\frac{1}{3}\cdot 64000-5\cdot 1600+2000\cdot 40-350\approx\boxed{50\,316{.}67\ \text{€}}.$