1. 15.7 Maximum and Minimum Values
MAT 3238 Vector Calculus
15.7 Maximum and Minimum Values

2. Homework
Both written and WA HW due Thursday.

3. One Variable Vs Two variables

4. Local Maximum

5. Critical Numbers, Critical Points𝑦 𝑥
𝑦=𝑓 𝑥

6. Fermat’s Theorem 𝑦 𝑥
𝑦=𝑓 𝑥

7. The Second Derivative Test

8. The Second Derivative Test𝑦
𝑦=𝑓 𝑥 𝑥 𝑐
Concave up at 𝑥=𝑐

9. Quote
When You Realize You’ve Hit Rock Bottom, There’s Only One Way To Go, And That’s Up!

10. Quote
When You Realize You’ve Hit Rock Bottom, There’s Only One Way To Go, And That’s Up! But along which Direction?

11. The Second Derivative Test𝑦
𝑦=𝑓 𝑥 𝑥 𝑐
Concave up at 𝑥=𝑐

12. The Second Derivative Test

13. The Second Derivative Test
Possible LMM or Inflection Point

14. The Second Derivative Test

15. Example 1

16. Example 1

17. SageMath Codes
var('x,y') plot3d(x*y*(1-x-y),(x,-1,1),(y,-1,1))

18. Absolute Minimum, Maximum
𝑦=𝑓(𝑥) on [𝑎,𝑏]
Closed Interval Method 𝑦
𝑦=𝑓 𝑥 𝑥 𝑏 𝑎

19. Absolute Minimum, Maximum
𝑧=𝑓(𝑥,𝑦) on a closed and bounded region 𝐷

20. Example 2

21. Example 2

22. Example 2 1 𝑦 𝑥
𝐿 1
𝐿 2
𝐿 3
𝐿 4 𝐷

23. Example 2 1 𝑦 𝑥
𝐿 1
𝐿 2
𝐿 3
𝐿 4 𝐷

24. Example 2 1 𝑦 𝑥
𝐿 1
𝐿 2
𝐿 3
𝐿 4 𝐷

25. Example 2 1 𝑦 𝑥
𝐿 1
𝐿 2
𝐿 3
𝐿 4 𝐷

26. Example 2 1 𝑦 𝑥
𝐿 1
𝐿 2
𝐿 3
𝐿 4 𝐷

27. -End- For Spring 2018

28. Things to Do Next Year
Embed a SageMath Webpage

29. Diagrams 𝑎 𝐼 𝑦 𝑥
𝑦=𝑓 𝑥

30. Diagrams 1 𝑦 𝑥
𝐿 1
𝐿 2
𝐿 3
𝐿 4 𝐷