# What is the minimum value of the function y=sqrt(x²+2ax+10a²) where a > 0 ?

In order to find the minimum value of the function we have to minimize $g \left(x\right) = {x}^{2} + 2 a x + 10 {a}^{2}$ which occurs at
$g ' \left(x\right) = 0 \implies 2 x + 2 a = 0 \implies x = - a$
and $g \left(- a\right) = 9 {a}^{2}$
Hence $f \left(- a\right) = \sqrt{g \left(- a\right)} = \sqrt{9 {a}^{2}} = 3 \left\mid a \right\mid = 3 a$ since $a > 0$