MAT 1234 Calculus I

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

2. Next • WebAssign 3.1 • Quiz– 2.7, 2.9

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) f has an absolute maximum at x=c on D if for all x in D (D =Domain of f) c D

10. Definition (Absolute Min) f has an absolute minimum at x=c on D if for all x in D (D =Domain of f) c D

11. Definition • The absolute maximum and minimum values of f are called the extreme values of f.

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

13. Definition (Local Max/Min) f has an local maximum at x=c if for all x in some open interval containing c f has an local minimum at x=c if for all x in some open interval containing c

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

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

16. The Extreme Value Theorem • If f is continuous on a closed interval [a,b], then f attains an absolute max value f(c) and an absolute min value f(d) at some numbers c and d in [a,b]. • No guarantee of absolute max/min if one of the 2 conditions are missing.

17. The Extreme Value Theorem • If f is continuous on a closed interval [a,b], then f attains an absolute max value f(c) and an absolute min value f(d) at some numbers c and d in [a,b]. • No guarantee of absolute max/min if one of the 2 conditions are missing.

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) • f is not continuous on [a,b] y y=f(x) x b a c

20. Example 2 (No abs. max/min) • The interval is not closed y y=f(x) x b a

21. How to find Absolute Max./Min.? • The Extreme Value Theoremguarantee of absolute max/min if f is continuous on a closed interval [a,b]. • Next: How to find them?

22. Fermat’s Theorem • If f has a local maximum or minimum at c, and if exists, then y c x

23. Q&A: T or F • The converse of the theorem: If , then f has a local maximum or minimum at c

24. Definition (Critical Number) • A critical numberof a function f is a number c in the domain of f such that either or does not exist.

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

26. Critical Number (Translation) • If the function is differentiable, critical points are those c such that

27. Example 3 • Find the critical numbers of

28. Example 3 • Find the critical numbers of

29. The Closed Interval Method • Idea: the absolute max/min values of a continuous function f on a closed interval [a,b] only occur at • the local max/min (the critical numbers) • end points of the interval

30. The Closed Interval Method • To find the absolute max/min values of a continuous function f on a closed interval [a,b]: • Find the values of f at the critical numbers of f in (a,b). • Find the values of f at the end points. • The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of the those values from is the absolute minimum value.

31. The Closed Interval Method • To find the absolute max/min values of a continuous function f on a closed interval [a,b]: • Find the values of f at the critical numbers of f in (a,b). • Find the values of f at the end points. • The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of the those values from is the absolute minimum value.

32. The Closed Interval Method • To find the absolute max/min values of a continuous function f on a closed interval [a,b]: • Find the values of f at the critical numbers of f in (a,b). • Find the values of f at the end points. • The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of the those values from is the absolute minimum value.

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

34. Expectations: Formal Conclusion

35. Classwork • Do part (a) only.