Minimum average cost with a hyperbolic term

Statement

The average production cost is $\bar C(q)=\dfrac{400}{q}+2q$ (in €), with $q>0$. Find the quantity that minimizes the average cost and the minimum value.

Solution

$\bar C'(q)=-\dfrac{400}{q^{2}}+2=0\Rightarrow q^{2}=200\Rightarrow q=\sqrt{200}\approx 14{.}14$. The minimum is $\bar C(\sqrt{200})=\frac{400}{\sqrt{200}}+2\sqrt{200}=2\sqrt{400\cdot 2}=2\sqrt{800}\approx\boxed{56{.}57\ \text{€}}.$