Mean Value of Concave Real Function

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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 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:

$\displaystyle \exists \eta \in \left({x \,.\,.\, \xi}\right): f^{\prime} \left({\eta}\right) = \dfrac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$

From Real Function is Concave iff Derivative is Decreasing, the derivative of $f$ is decreasing.

Thus:

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

Hence:

$f \left({x}\right) - f \left({\xi}\right) \le f' \left({\xi}\right) \left({x - \xi}\right)$

$\blacksquare$


Also see