## Rolle's theorem---an application?

Let $f$ be twice differentiable on $[a,b]$, and $|f(x)|\leq 1,\ x\in [-2,2]$. Assume that $\frac{1}{2}[f'(0)]^2+f^3(0)>\frac{3}{2}$. Prove that there exists an $x_0\iin (-2,2)$ such that $f''(x_0)+3f^2(x_0)=0.$

Would you help me? I could not find any fine property of $F(x)=\frac{1}{2}[f'(x)]^2+f^3(x).$