> 
> Let $f(x)$ be twice differentiable in $(a,b)$ and suppose that $f'(a)=f'(b)=0$. Prove that there exist at least one point $\mu$ in $(a,b)$ such that
>  $$
> |f''(\mu)|\geq\frac{4}{(b-a)^2}[f(b)-f(a)].
> $$
> 
> 
> 

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I tried Rolle's theorem. Since $f'(a)=f'(b)=0$, there exists at least one $\mu \in (a,b)$ such that
$$
f''(\mu)=0.
$$
I also wrote the right hand side as
$$
\dfrac{f(b)-f(a)}{(\dfrac{b-a}{2})^2}=\dfrac{\dfrac{f(b)-f(\dfrac{b+a}{2})+f(\dfrac{b+a}{2})-f(a)}{\dfrac{b-a}{2}}}{\dfrac{b-a}{2}}
$$
But I do not know how to do next.