For differentiable functions, convexity has useful local tests.

A differentiable function $f$ on a convex domain is convex exactly when every tangent hyperplane is a global under-estimator:

$$f(y)\ge f(x)+\nabla f(x)^T(y-x)$$

for all $x$ and $y$ in the domain.

If $f$ is twice differentiable, convexity can instead be checked through curvature: the Hessian must be positive semidefinite at every point of the convex domain.

At a stationary point $x_*$ of a differentiable convex function, the first-order inequality gives

$$f(y)\ge f(x_*)$$

for every feasible $y$. A stationary point is therefore a global minimizer.

These criteria connect the global geometry of convexity with the local derivative information used by optimization algorithms.