$f(x)$ is continuously differentiable on $[0,1]$, prove that $ (\int_0^1 f(x)dx)^2 \leq \frac {1} {12} \int_0^1 (f'(x))^2 dx $

Question

$f(x)$ is a real function continuously differentiable on $[0,1]$. $f(0) = f(1) = 0$

Prove that $$ \left(\int_0^1 f(x)\mathrm{dx}\right)^2 \leq \frac {1} {12} \int_0^1 (f'(x))^2 \mathrm{dx} $$