saddle point

A critical point that curves upward in one direction and downward in another, with no single-variable analogue.

A saddle point is a critical point where the surface curves up in one direction and down in another at the very same spot — a third fate for a critical point beyond local max or local min, one with no counterpart in single-variable calculus. The second derivative test flags a saddle whenever the discriminant $D<0$ , regardless of the sign of $f_{xx}$ , since there is no single concavity direction to check.

Examples

For $f(x,y)=x^2-y^2$ at $(0,0)$ : $D=(2)(-2)-0^2=-4<0$ , so $(0,0)$ is a saddle — the surface rises along the $x$ -axis and falls along the $y$ -axis.
Of the two critical points of $f(x,y)=x^3-3xy+y^3$ , $(0,0)$ turns out to be a saddle point while $(1,1)$ is a local minimum.