Difference between revisions of "Math 22 Extrema of Functions of Two Variables"

From Math Wiki
Jump to navigation Jump to search
 
(One intermediate revision by the same user not shown)
Line 16: Line 16:
 
   <math>f_x(x_0,y_0)=0</math> and <math>f_y(x_0,y_0)=0</math>
 
   <math>f_x(x_0,y_0)=0</math> and <math>f_y(x_0,y_0)=0</math>
  
−
'''Example:''' Find relative extrema of:
+
'''Example:''' Find the relative critical point of of:
  
 
'''1)''' <math>f(x,y)=2x^2+y^2+8x-6y+20</math>
 
'''1)''' <math>f(x,y)=2x^2+y^2+8x-6y+20</math>
Line 26: Line 26:
 
|and: <math>f_y(x,y)=2y-6=0</math>, so <math>y=3</math>
 
|and: <math>f_y(x,y)=2y-6=0</math>, so <math>y=3</math>
 
|-
 
|-
−
|Therefore, there is a relative extrema at <math>(-2,3)</math>
+
|Therefore, there is a critical point at <math>(-2,3)</math>
 
|}
 
|}
  
Line 32: Line 32:
 
   Let <math>f</math> have continuous second partial derivatives on an open region containing <math>(a,b)</math> for which <math>f_x(a,b)=0</math> and <math>f_y(a,b)=0</math>
 
   Let <math>f</math> have continuous second partial derivatives on an open region containing <math>(a,b)</math> for which <math>f_x(a,b)=0</math> and <math>f_y(a,b)=0</math>
 
   Then, consider <math>d=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2</math>
 
   Then, consider <math>d=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2</math>
 +
 
 +
  Then:
 +
  1. If <math>d>0</math> and <math>f_{xx}(a,b)>0</math>, then <math>f</math> has a relative minimum at <math>(a,b)</math>.
 +
  2. If <math>d>0</math> and <math>f_{xx}(a,b)<0</math>, then <math>f</math> has a relative maximum at <math>(a,b)</math>.
 +
  3. If <math>d<0</math>, then <math>(a,b,f(a,b))</math> is a saddle point.
 +
  4. If <math>d=0</math>, no conclusion.
 +
 +
'''Example:''' Find the relative extrema (maximum or minimum):
 +
 +
'''1)''' <math>f(x,y)=2x^2+y^2+8x-6y+20</math>
 +
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|Consider: <math>f_x(x,y)=4x+8=0</math>, so <math>x=-2</math>
 +
|-
 +
|and: <math>f_y(x,y)=2y-6=0</math>, so <math>y=3</math>
 +
|-
 +
|Therefore, there is a critical point at <math>(-2,3)</math>
 +
|-
 +
|Now: <math>f_{xx}f(x,y)=4</math>
 +
|-
 +
|<math>f_{yy}f(x,y)=2</math>
 +
|-
 +
|and <math>f_{xy}f(x,y)=0</math>
 +
|-
 +
|Then, <math>d=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2=(4)(2)-0^2=8</math>
 +
|-
 +
|Since, <math>d>0</math> and <math>f_{xx}f(x,y)=4>0</math>, then by the second-partial test, <math>f</math> has a relative minumum at <math>(-2,3)</math>
 +
|}
  
 
[[Math_22| '''Return to Topics Page''']]
 
[[Math_22| '''Return to Topics Page''']]
  
 
'''This page were made by [[Contributors|Tri Phan]]'''
 
'''This page were made by [[Contributors|Tri Phan]]'''

Latest revision as of 08:32, 18 August 2020

Relative Extrema of a Function of Two Variables

 Let  be a function defined on a region containing . The function  has a relative maximum at  when there is a circular region  centered at  such that
 
 
 
 for all  in .

 The function  has a relative minimum at  when there is a circular region  centered at  such that
 
 
 
 for all  in .

First-Partials Test for Relative Extrema

 If  has a relative extremum at  on an open region  in the xy-plane, and the first partial derivatives of  exist in , then
 
  and

Example: Find the relative critical point of of:

1)

Solution:  
Consider: , so
and: , so
Therefore, there is a critical point at

The Second-Partials Test for Relative Extrema

 Let  have continuous second partial derivatives on an open region containing  for which  and 
 Then, consider 
 
 Then:
 1. If  and , then  has a relative minimum at .
 2. If  and , then  has a relative maximum at .
 3. If , then  is a saddle point.
 4. If , no conclusion.

Example: Find the relative extrema (maximum or minimum):

1)

Solution:  
Consider: , so
and: , so
Therefore, there is a critical point at
Now:
and
Then,
Since, and , then by the second-partial test, has a relative minumum at

Return to Topics Page

This page were made by Tri Phan

Retrieved from "https://wiki.math.ucr.edu/index.php?title=Math_22_Extrema_of_Functions_of_Two_Variables&oldid=2601"