$f(x)$ 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 $
How prove this inequality $\left(\int_{0}^{1}f(x)dx\right)^2\le\frac{1}{12}\int_{0}^{1}|f'(x)|^2dx$