RO-NMO 2008 Problem 122

Let $f: [0,1]\to\mathbb R$ be a derivable function, with a continuous derivative $f'$ on $[0,1]$ . Prove that if $f\left( \frac 12\right) = 0$ , then $$\int^1_0 \left( f'(x) \right)^2 dx \geq 12 \left( \int^1_0 f(x) dx \right)^2.$$