$13$

$(12)$ Let $u$:$[0,)R$ be differentiable function $($in particular $u^{\text{'}}(0+)$ exist$),$ so that $u(0)=0$ and $u^{\text{'}}(x)$:$(0,)R$ is continuous. Show the following "uncertainty" inequality:

${\displaystyle \underset{0}{\overset{x}{}}}(u^{\text{'}}(y){)}^{2}dy\frac{u^2(x)}{x}$

for every $x>0.$