Mean Value of Convex Real Function

Theorem
Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$ and differentiable on the open interval $\left({a \,.\,.\, b}\right)$.

Let $f$ be convex on $\left({a \,.\,.\, b}\right)$.

Then:
 * $\forall \xi \in \left({a \,.\,.\, b}\right): f \left({x}\right) - f \left({\xi}\right) \ge f^{\prime} \left({\xi}\right) \left({x - \xi}\right)$

Proof
By the Mean Value Theorem:
 * $\exists \eta \in \left({x \,.\,.\, \xi}\right): f' \left({\eta}\right) = \dfrac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$

From Derivative of Convex Real Function is Increasing, the derivative of $f$ is increasing.

Thus:
 * $x > \xi \implies f' \left({\eta}\right) \ge f' \left({\xi}\right)$
 * $x < \xi \implies f' \left({\eta}\right) \le f' \left({\xi}\right)$

Hence:
 * $f \left({x}\right) - f \left({\xi}\right) \ge f' \left({\xi}\right) \left({x - \xi}\right)$

Also see

 * Mean Value of Concave Real Function