Section 4.7 - Maxima and Minima of Functions of Two Variables

Section Objectives

  1. Find the critical points of a function of two variables.
  2. Use the second partials test to classify critical points.
  3. Use critical points and boundary points to find the absolute extrema for functions of two variables.



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.


Critical points

Definition:

Suppose is defined on an open region containing . The point is a critical point of if

  1. or
  2. does not exist.


Examples







Extreme values

Critical points are useful for finding extreme values.

Definition:

Suppose is defined on a region containing .

  1. is an absolute (global) maximum value if for all in .
  2. is an absolute (global) minimum value if for all in .


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 .

  1. is a relative (local) maximum value if for all in some open disk containing .
  2. is a relative (local) minimum value if for all in some open disk 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 .



Second partials test

Theorem:

Suppose has continuous second partial derivatives in an open region containing the point for which . Define the number by

  1. If and , then has a relative minimum at .
  2. If and , then has a relative maximum at .
  3. If , then has a saddle point at .
  4. If , then this test is inconclusive.


Examples