1. 13.8 Extrema of Functions of two variables

5. Warning: extrema occur only at critical points but not all critical points are extrema!!!

6. Example 1
Algebraically determine the relative extrema of
(-2,3,3)

7. Example 1 solution

8. Example 2
Algebraically determine the relative extrema of
(0,0,1)

9. Example 2 solution

10. Warning: Not all critical points are maximums or minimums
Warning: Not all critical points are maximums or minimums. This is called a saddle point. (it is a maximum in the x direction and a minimum in the y direction.

12. Example 3
Use the Second partials test to find and classify the relative extrema of

13. Example 3 Solution

14. Example 3 diagram
(4/3,4/3,2.185)
(0,0,1)

15. Example 4
Use the Second partials test to find and classify the relative extrema of

16. Example 4 solution

17. Example 4 Diagram
(0,0,0)

18. A few years ago,It was reported on CNN that a new prime number was discovered that is four times larger than the largest prime number that was previously known.
What is wrong with this statement?