= Second-derivative criterion for convexity
{title2=$f\text{ convex}\iff f''\ge0$}
A twice-differentiable <real-valued function> on an open <interval> is <convex> if and only if its <second derivative> is nonnegative everywhere. If $f''\ge0$, the <mean value theorem> makes $f'$ nondecreasing, which implies the supporting-line and chord inequalities for <convexity>. Conversely, the slopes of a <convex function> are ordered; where the <second derivative> exists, this gives $f''\ge0$. Strict positivity of $f''$ is sufficient for <strict convexity>, but is not necessary: $f(x)=x^4$ is <strictly convex> despite $f''(0)=0$.
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