Suppose $\displaystyle f(x)$ is a diferentiable function and $\displaystyle g(x)$ is a doubly differentiable function such that $\displaystyle f'(x)=g(x)$ and $\displaystyle |f(x)| \leq 1$ and $\displaystyle |g(x)| \geq 2$ for all $\displaystyle x \in [-3,3]$. If further $\displaystyle [f(0)]^{2}+[g(0)]^{2}=9$, prove that there exists some $\displaystyle c \in (-3,3)$ such that $\displaystyle g(c) \cdot g''(c)<0$.