Second Derivative Test

The second derivative test allows one to figure out if a critical point is a local minimum, local maximum, or saddle point. It uses partial derivatives and is defined as the following:

1. If and , then has a local minimum at .
2. If and , then has a local maximum at .
3. If , then has a saddle point at .
4. If , then no conclusion can be drawn.

See Multivariable Calculus Notes - Chapter 14 - Sections 7-8 for source, more information, and examples.