Minimum average cost

Statement

The total cost is $C(q)=q^{3}-5q^{2}+60q$ (in €). Find the quantity that minimizes the average cost $\dfrac{C(q)}{q}$ and the minimum value.

Solution

$AC(q)=\dfrac{C(q)}{q}=q^{2}-5q+60$. Then $AC'(q)=2q-5=0\Rightarrow q=2{.}5$; $AC''=2>0$: minimum. $AC(2{.}5)=2{.}5^{2}-5\cdot 2{.}5+60=\boxed{53{.}75\ \text{€}}.$