Suppose $f(x)$ is a diferentiable function and $g(x)$ is a doubly differentiable function such that $f'(x)=g(x)$ and $|f(x)| \leq 1$ and $|g(x)| \geq 2$ for all $x \in [-3,3]$. If further $[f(0)]^{2}+[g(0)]^{2}=9$, prove that there exists some $c \in (-3,3)$ such that $g(c) \cdot g''(c)<0$.
There is no condition on $g$ as $|g(x)|>2$.