Section 12.7 Maxima and Minima of Functions
A function f ( x , y ) has a local max at a point ( a, b ) if there is a disk centered at ( a, b ) with f ( x , y )
 f ( a , b ) for all ( x,y ) in the disk.
Similarly, A function
( a, b ) with f ( x , y )
 f ( x , y ) has a local min at a point ( a, b ) if there is a disk centered at f ( a , b ) for all ( x,y ) in the disk.
Thm.
If a then function, f x
( a , b ) f ( x , y ), has a local and f y
( a , b ) are max both or equal min at ( a , b ) and if f x
( a , b ) and
to zero.
f y
( a , b ) exist,
Definition A critical point is a point at which both first partial derivatives are zero.
It need not be a local extreme point (max or min) to be a critical point.
Example 1 Find the critical points of f ( x , y )

3 x
2 
4 y
2 
8 xy

2 x

6 y .
Example 2 Show that f ( x , y )
 xe xy
has no local extreme points.
After finding all critical points, we apply the 2nd derivative test to each one to see if it is a local max or a local min or a saddle point. If this test fails, we go to a computer or some other method not done in this course.
The 2nd Derivative Test If all 2nd partial derivatives of f ( x , y ) exist and are continuous and if both
D ( a , b )
 f x
( a , b ) f xx
( a , b ) and f y
( a , b ) f yy
( a , b )

 f xy
are equal to 0, then find
( a , b )

2 case 1
If D ( a,b ) > 0 then f has an extreme value at ( a,b ). In this case, f xx and f yy
will have the same sign and be nonzero.
If f xx and f yy
are both positive, the extreme value is a local min.
If f xx and f yy
are both negative, the extreme value is a local max. case 2
If D ( a,b ) < 0 then f has a saddle point at ( a,b ). case 3
If D ( a,b ) = 0 the test is inconclusive.
Example 3
Test the critical point of example 1, above.
More examples are at the end of these notes.

Absolute max and min A function with domain D has an absolute max at ( a,b ) in D if f ( x , y )
 f ( a , b ) for all ( x,y ) in D. Similarly, it has an absolute min at ( a,b ) if f ( x , y )
 f ( a , b )
.
Thm. A continuous function on a closed bounded domain D must attain an absolute max and an absolute min on D. These must occur at local extreme points or on the boundary of D.
Examples:
1. f ( x , y )

2 x
2 
5 y
2 
6 xy

6 x

14 y
2. f ( x , y )
 
27 x
3  y
3 
81 xy
3. f ( x , y )

3 x
2 y

4 y
3 
3 x
2
4. Problem 50 pg 782 in Stewart. An aquarium has rectangular base and sides. The volume, V, is given. the base is made of slate and the sides are made of glass. If the cost per unit area for slate is five times that of glass, find the dimensions that minimize the cost.
Check out problems 33 and 34.