Mean Value of Convex and Concave Functions

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:

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

$\blacksquare$

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:

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

$\blacksquare$

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 12.21 \ (4)$