second derivative test

The two-variable test using $D=f_{xx}f_{yy}-f_{xy}^2$ to classify a critical point as a local max, local min, saddle, or inconclusive.

The second derivative test classifies a critical point $(a,b)$ using the discriminant $D=f_{xx}f_{yy}-f_{xy}^2$ evaluated there. When $D>0$ , $f_{xx}$ and $f_{yy}$ are forced to share the same sign, so checking $f_{xx}$ alone settles the case: $f_{xx}>0$ gives a local minimum, $f_{xx}<0$ a local maximum. When $D<0$ , the point is a saddle point regardless of $f_{xx}$ 's sign. When $D=0$ the test is inconclusive — the quadratic approximation it relies on isn't sharp enough to decide, and direct comparison of nearby function values is the fallback.

Examples

At $(1,1)$ for $f(x,y)=x^3-3xy+y^3$ : $f_{xx}=6x$ , $f_{yy}=6y$ , $f_{xy}=-3$ , so $D=(6)(6)-(-3)^2=27>0$ and $f_{xx}=6>0$ , a local minimum.