Let $f: S \to \mathbb{R}$ be twice continuously differentiable on a convex open set $S \subseteq \mathbb{R}^n$. Then $f$ is convex on $S$ if and only if the Hessian matrix \begin{align*} (H_f)_{ij}(x) = \frac{\partial^2 f}{\partial x_i \partial x_j}(x) \end{align*} is positive semidefinite for every $x \in S$.