Mean value theorem for convex functions Let $f$ be a real function with left and right derivatives $f'_-$ and $f'_+$ on the open interval $(a,b)$, and continuous on $[a,b]$ (e.g., let $f$ be convex on $[a,b]$). Then,
Is there something like the mean value theorem for $f$?

By the mean value version of Taylor's theorem we have:
\begin{align}
f(y) &=f(x)+f'(x)(y-x)+\dfrac{1}{2}f''(z)(y-x)^2, \text{for some }z\in[x,y].\\
\end{align}
