At a stationary point $x_*$, the Hessian reveals the local curvature that the zero gradient alone cannot distinguish.

If $H_f(x_)$ is positive definite, then $x_$ is a strict local minimum. If the Hessian is negative definite, the point is a strict local maximum. If it is indefinite, the point is a saddle.

A positive-semidefinite or negative-semidefinite Hessian may be inconclusive because higher-order terms can determine the local behavior.

The second-order test therefore combines two pieces of information: the gradient must vanish, and the Hessian must curve consistently in the relevant directions.

First-order conditions locate candidates; second-order curvature distinguishes the usual local geometries around those candidates.