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

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