Section Objectives
For functions of a single variable, critical points were domain interior points at which the derivative was zero or not defined. These points gave the locations of possible extreme values. For two-variable functions, we will see similar ideas.
Definition:
Suppose is defined on an open region containing . The point is a critical point of if
Critical points are useful for finding extreme values.
Definition:
Suppose is defined on a region containing .
Theorem:
Suppose is continuous on the closed and bounded region in the -plane. Then takes on both an absolute maximum value and an absolute minimum value on .
Definition:
Suppose is defined on a region containing .
Theorem:
If has a relative extreme value at on an open region , then is a critical point of .
Definition:
Suppose is defined at the point and . The point is called a saddle point if but does not have a maximum or minimum value at .
Theorem:
Suppose has continuous second partial derivatives in an open region containing the point for which . Define the number by