Let ${\rm f}:\left[a, b\right]\to\mathbb{R}$ be twice differentiable, and suppose 

$$\lim\_{x\to a^{+}}
\frac{{\rm f}\left(x\right) - {\rm f}\left(a\right)}{x - a}
=
\lim\_{x\to b^{-}}\frac{{\rm f}\left(x\right) - {\rm f}\left(b\right)}{x - b}
=0
$$

Show that there exists $\xi \in \left(a, b\right)$ such that
$\displaystyle{%
\left\vert\vphantom{\Large A}\,{\rm f}''\left(\xi\right)\right\vert
\geq
\frac{4\left\vert\vphantom{\Large A}%
\,{\rm f}\left(a\right) - {\rm f}\left(b\right)\right\vert}
{\left(b - a\right)^{2}}}$.

I don't know how to start. Any hints ?.