Real Function is Convex iff Derivative is Increasing

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

Then $f$ is convex on $\left({a \,.\,.\, b}\right)$ iff its derivative $f'$ is increasing on $\left({a \,.\,.\, b}\right)$.

Thus the intuitive result that a convex function "gets steeper".

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

Let $x_1, x_2, x_3, x_4 \in \left({a \,.\,.\, b}\right): x_1 < x_2 < x_3 < x_4$.

By the definition of convex function:


 * $\dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \le \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2} \le \dfrac {f \left({x_4}\right) - f \left({x_3}\right)} {x_4 - x_3}$

Ignore the middle term and let $x_2 \to x_1^+$ and $x_3 \to x_4^-$.

Thus:
 * $f' \left({x_1}\right) \le f' \left({x_4}\right)$

Hence $f'$ is increasing on $\left({a \,.\,.\, b}\right)$.

Sufficient Condition
Let $f'$ be increasing on $\left({a \,.\,.\, b}\right)$.

Let $x_1, x_2, x_3 \in \left({a \,.\,.\, b}\right): x_1 < x_2 < x_3$.

By the Mean Value Theorem:


 * $\exists \xi: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} = f^{\prime} \left({\xi}\right)$
 * $\exists \eta: \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2} = f^{\prime} \left({\eta}\right)$

where $x_1 < \xi < x_2 < \eta < x_3$.

Since $f'$ is increasing:
 * $f' \left({\xi}\right) \le f' \left({\eta}\right)$

Thus:
 * $\dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \le \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$

Hence $f$ is convex by definition.

Also see

 * Derivative of Strictly Convex Real Function is Strictly Increasing


 * Derivative of Concave Real Function is Decreasing
 * Derivative of Strictly Concave Real Function is Strictly Decreasing