Convexity characterized by monotonicity of the derivative

Theorem. Let $f:I\to\mathbb{R}$ be differentiable on a nonempty open interval $I\subset\mathbb{R}$. Then $f$ is convex on $I$ if and only if $f'$ is nondecreasing on $I$.

Context. This provides a practical test for convexity in one variable and connects geometric convexity with calculus.

Proof sketch.

• If $f$ is convex, apply slope inequalities and take limits as $x\downarrow a$ and $x\uparrow b$ to get $f'(a)\le f'(b)$ for $a<b$.
• If $f'$ is nondecreasing, then for $a<b$ the mean value theorem implies the secant slope $(f(b)-f(a))/(b-a)$ lies between values of $f'$, giving Jensen's inequality and hence convexity.