Minimum average cost from a cubic

Statement

The total cost is $C(q)=q^{3}-6q^{2}+30q$ (in €). Find the quantity that minimizes the average cost $\dfrac{C(q)}{q}$ and the minimum value.

Solution

$AC(q)=q^{2}-6q+30$; $AC'(q)=2q-6=0\Rightarrow q=3$; $AC''=2>0$: minimum. $AC(3)=9-18+30=\boxed{21\ \text{€}}.$