Critical Points, Saddle Points, and the Second Derivative Test

The first step in any multivariate optimisation — portfolio construction, model calibration, MLE — is finding where the gradient vanishes: the critical points of the objective function. But a critical point may be a local minimum (what we want), a local maximum (often what we want to avoid), or a saddle point (neither — a direction of ascent and a direction of descent both exist). The second derivative test uses the discriminant $D = f_{xx}f_{yy} - f_{xy}^2$ to classify all three cases for $f\colon\mathbb{R}^2\to\mathbb{R}$ without having to examine every direction. This note develops the test from first principles, works through the geometry of the saddle point, and connects the three cases to the curvature of the objective surface.

Learning objectives. By the end of this note you should be able to: