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.