12.4 Local and global maximum and minimum

1. Can you find and prove an analogue for Taylor's Theorem for the multi-variable function?

2. Using 1, prove Second partial derivative Test.

3-4. Locate the maxima, minima, saddle points of the functions

3. .

http://matrix.skku.ac.kr/cal-lab/cal-12-4-3.html

Sol

[[x == 0, y == 0]]

[ 0 1/2]

[1/2 0]

-1/4 Saddle point.

4.

5-6. Let be the unit disc in ,

5. Let be a function on satisfying . Show that cannot have a maximum point in the interior

of .

6. If is a solution of Delta on , then show that if achieves its maximum in the interior, then is

identically constant.

7. Let be a function on satisfying . Show that cannot have a minimum point in the interior

of .

8. If is a solution of on , then show that if achieves its minimum in the interior, then is

identically constant.