Convexity via nonnegative second derivative

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

Connection. This follows from convexity ⇔ monotone derivative, since $f''\ge 0$ is equivalent to $f'$ being nondecreasing.