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 $\left({a \,.\,.\, b}\right)$.

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

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

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

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

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

Proof for Convex Function
Let $f$ be convex.

Then its derivative is increasing.

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

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

Proof for Concave Function
Let $f$ be concave.

Then its derivative is decreasing.

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

Hence:
 * $f \left({x}\right) - f \left({\xi}\right) \le f^{\prime} \left({\xi}\right) \left({x - \xi}\right)$