Math 140

1. Math 140 3.1 – Increasing and Decreasing Functions; Relative Extrema

3. Let be defined on the interval . Suppose and are in that interval. is ___________________ on that interval if when is ___________________ on that interval if when

4. Let be defined on the interval . Suppose and are in that interval. is ___________________ on that interval if when is ___________________ on that interval if when increasing

5. Let be defined on the interval . Suppose and are in that interval. is ___________________ on that interval if when is ___________________ on that interval if when increasing decreasing

6. Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE)

7. Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE)

8. Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE) In other words, where might change signs?

9. Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE) In other words, where might change signs?

10. Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE) In other words, where might change signs?

11. 1. When

12. 1. When

13. 1. When

14. 1. When

15. 1. When

16. 2. When does not exist (DNE)

17. 2. When does not exist (DNE)

18. 2. When does not exist (DNE)

19. Ex 1. Find the intervals of increase and decrease for

20. Ex 1. Find the intervals of increase and decrease for

21. Relative Extrema

22. Relative Extrema

23. Relative Extrema

24. Relative Extrema

25. Relative Extrema

26. Relative Extrema

27. If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur.

28. If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur. critical number

29. If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur. critical number

30. If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur. critical number critical point

31. If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur. critical number critical point

32. First Derivative Test for Relative Extrema Let be a critical number. Relative max at

33. First Derivative Test for Relative Extrema Let be a critical number. Relative max at Relative min at

34. First Derivative Test for Relative Extrema Let be a critical number. Not a relative extremum

35. First Derivative Test for Relative Extrema Let be a critical number. Not a relative extremum Not a relative extremum

36. Ex 2. Find all critical numbers for , and classify each critical point as a relative maximum, relative minimum, or neither.

37. Ex 2. Find all critical numbers for , and classify each critical point as a relative maximum, relative minimum, or neither.

38. Ex 3. Find all critical numbers for , and classify each critical point as a relative maximum, relative minimum, or neither.

39. Ex 3. Find all critical numbers for , and classify each critical point as a relative maximum, relative minimum, or neither.