Study of a firm's maximum profit

Statement

Study the profit function $\pi(q)=-q^{3}-6q^{2}+1440q-500$ (in €) for $q>0$: find the profit-maximizing quantity and the maximum profit.

Solution

$\pi'(q)=-3q^{2}-12q+1440=0\Rightarrow q^{2}+4q-480=0\Rightarrow (q-20)(q+24)=0$. Discard $q=-24$; with $\pi''=-6q-12$ we get $\pi''(20)=-132<0$: maximum at $q=20$. $\pi(20)=-8000-2400+28800-500=\boxed{17\,900\ \text{€}}.$