A Saddle Point

Critical points of a function of two variables are those points at which both partial derivatives of the function are zero. A critical point of a function of a single variable is either a local maximum, a local minimum, or neither. With functions of two variables there is a fourth possibility - a saddle point.

A point is a saddle point of a function of two variables if

2 2 [ 2 ] 2 @f-= 0, @f--= 0, and @-f- @-f-- -@-f- < 0 @x @y @x2 @y2 @x @y

at the point.

The surface shown below is the graph of

2 2 f |\x,y/| = x - y .

It has a saddle point at the origin.