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).