 Second-derivative criterion for convexity

A twice-differentiable real-valued function on an open interval is convex if and only if its second derivative is nonnegative everywhere. If $f^{\prime \prime }\ge 0$, the mean value theorem makes $f^{\prime }$ 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 $f^{\prime \prime }\ge 0$. Strict positivity of $f^{\prime \prime }$ is sufficient for strict convexity, but is not necessary: $f(x)=x^4$ is strictly convex despite $f^{\prime \prime }(0)=0$.