1. MAT 1234 Calculus I Section 3.1 Maximum and Minimum Values http://myhome.spu.edu/lauw

2. Next WebAssign 3.1 Quiz– 2.8

3. 1 Minute… You can learn all the important concepts in 1 minute.

4. 1 Minute… High/low points – most of them are at points with horizontal tangent

5. 1 Minute… High/low points – most of them are at points with horizontal tangent. Highest/lowest points – at points with horizontal tangent or endpoints

6. 1 Minute… You can learn all the important concepts in 1 minute. We are going to develop the theory carefully so that it works for all the functions that we are interested in. There are a few definitions…

7. Preview Definitions absolute max/min local max/min critical number Theorems Extreme Value Theorem Fermat’s Theorem The Closed Interval Method

8. Max/Min We are interested in max/min values Minimize the production cost Maximize the profit Maximize the power output

9. Definition (Absolute Max) c D

10. Definition (Absolute Min) c D

11. Definition

12. x Example 1 y Absolute max. Absolute min.

13. Definition (Local Max/Min)

14. x Example 1 y Local max. Local min.

15. Q&A An end point is not a local max/min, why?

16. The Extreme Value Theorem

18. Q&A Give 2 examples of functions on an interval that do not have absolute max value.

19. Example 2 (No abs. max/min)

20. The interval is not closed

21. How to find Absolute Max./Min.?

22. Fermat’s Theorem c x y

23. Q&A: T or F

24. Definition (Critical Number)

25. Critical Number (Translation) Critical numbers give all the potential local max/min values

26. Critical Number (Translation)

27. Example 3 Find the critical numbers of

28. Example 3 Find the critical numbers of

29. The Closed Interval Method

33. Example 4 Find the absolute max/min values of

34. Expectations: Formal Conclusion