Integral Inequality from unit interval to reals with $f$ and $f'$

Question

For continuously differentiable function $f : [0, 1] → R$ with $ f \big( \frac{1}{2} \big) = 0$, show that $$ \left(\int_0^1 f(x) dx\right)^{2} \leq \frac{1}{4} \int_0^1 (f' (x))^{2}dx $$