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

3. Let 𝑓(𝑥) be defined on the interval 𝑎<𝑥<𝑏
Let 𝑓(𝑥) be defined on the interval 𝑎<𝑥<𝑏. Suppose 𝑥 1 and 𝑥 2 are in that interval. 𝑓 is ___________________ on that interval if 𝑓 𝑥 2 >𝑓( 𝑥 1 ) when 𝑥 2 > 𝑥 1 𝑓 is ___________________ on that interval if 𝑓 𝑥 2 <𝑓( 𝑥 1 ) when 𝑥 2 > 𝑥 1

4. Let 𝑓(𝑥) be defined on the interval 𝑎<𝑥<𝑏
Let 𝑓(𝑥) be defined on the interval 𝑎<𝑥<𝑏. Suppose 𝑥 1 and 𝑥 2 are in that interval. 𝑓 is ___________________ on that interval if 𝑓 𝑥 2 >𝑓( 𝑥 1 ) when 𝑥 2 > 𝑥 1 𝑓 is ___________________ on that interval if 𝑓 𝑥 2 <𝑓( 𝑥 1 ) when 𝑥 2 > 𝑥 1
increasing

5. Let 𝑓(𝑥) be defined on the interval 𝑎<𝑥<𝑏
Let 𝑓(𝑥) be defined on the interval 𝑎<𝑥<𝑏. Suppose 𝑥 1 and 𝑥 2 are in that interval. 𝑓 is ___________________ on that interval if 𝑓 𝑥 2 >𝑓( 𝑥 1 ) when 𝑥 2 > 𝑥 1 𝑓 is ___________________ on that interval if 𝑓 𝑥 2 <𝑓( 𝑥 1 ) when 𝑥 2 > 𝑥 1
increasing
decreasing

6. Recall: 𝑓 is increasing where 𝑓 ′ 𝑥 >0, and decreasing where 𝑓 ′ 𝑥 <0. Where might 𝑓(𝑥) change from increasing to decreasing, or decreasing to increasing? 1. When 𝑓 ′ 𝑥 =0 2. When 𝑓′(𝑥) does not exist (DNE)

7. Recall: 𝑓 is increasing where 𝑓 ′ 𝑥 >0, and decreasing where 𝑓 ′ 𝑥 <0. Where might 𝑓(𝑥) change from increasing to decreasing, or decreasing to increasing? 1. When 𝑓 ′ 𝑥 =0 2. When 𝑓′(𝑥) does not exist (DNE)

8. Recall: 𝑓 is increasing where 𝑓 ′ 𝑥 >0, and decreasing where 𝑓 ′ 𝑥 <0. Where might 𝑓(𝑥) change from increasing to decreasing, or decreasing to increasing? 1. When 𝑓 ′ 𝑥 =0 2. When 𝑓′(𝑥) does not exist (DNE)
In other words,
where might
𝑓 ′ 𝑥 change
signs?

9. Recall: 𝑓 is increasing where 𝑓 ′ 𝑥 >0, and decreasing where 𝑓 ′ 𝑥 <0. Where might 𝑓(𝑥) change from increasing to decreasing, or decreasing to increasing? 1. When 𝒇 ′ 𝒙 =𝟎 2. When 𝑓′(𝑥) does not exist (DNE)
In other words,
where might
𝑓 ′ 𝑥 change
signs?

10. Recall: 𝑓 is increasing where 𝑓 ′ 𝑥 >0, and decreasing where 𝑓 ′ 𝑥 <0. Where might 𝑓(𝑥) change from increasing to decreasing, or decreasing to increasing? 1. When 𝒇 ′ 𝒙 =𝟎 2. When 𝒇′(𝒙) does not exist (DNE)
In other words,
where might
𝑓 ′ 𝑥 change
signs?

11. 1. When 𝑓 ′ 𝑥 =0

12. 1. When 𝑓 ′ 𝑥 =0

13. 1. When 𝑓 ′ 𝑥 =0

14. 1. When 𝑓 ′ 𝑥 =0

15. 1. When 𝑓 ′ 𝑥 =0

16. 2. When 𝑓′(𝑥) does not exist (DNE)

17. 2. When 𝑓′(𝑥) does not exist (DNE)

18. 2. When 𝑓′(𝑥) does not exist (DNE)

19. Ex 1. Find the intervals of increase and decrease for 𝑓 𝑥 = 𝑥 2 𝑥−2

20. Ex 1. Find the intervals of increase and decrease for 𝑓 𝑥 = 𝑥 2 𝑥−2

21. Relative Extrema

22. Relative Extrema

23. Relative Extrema

24. Relative Extrema

25. Relative Extrema

26. Relative Extrema

27. If 𝑓(𝑐) is defined, and either 𝑓 ′ 𝑐 =0 or 𝑓′(𝑐) DNE, then 𝑥=𝑐 is called a ________________. Also, 𝑐, 𝑓 𝑐 is called a __________________. Critical numbers give us 𝑥-values where relative extrema might occur.

28. If 𝑓(𝑐) is defined, and either 𝑓 ′ 𝑐 =0 or 𝑓′(𝑐) DNE, then 𝑥=𝑐 is called a ________________. Also, 𝑐, 𝑓 𝑐 is called a __________________. Critical numbers give us 𝑥-values where relative extrema might occur.
critical number

29. If 𝑓(𝑐) is defined, and either 𝑓 ′ 𝑐 =0 or 𝑓′(𝑐) DNE, then 𝑥=𝑐 is called a ________________. Also, 𝑐, 𝑓 𝑐 is called a __________________. Critical numbers give us 𝑥-values where relative extrema might occur.
critical number

30. If 𝑓(𝑐) is defined, and either 𝑓 ′ 𝑐 =0 or 𝑓′(𝑐) DNE, then 𝑥=𝑐 is called a ________________. Also, 𝑐, 𝑓 𝑐 is called a __________________. Critical numbers give us 𝑥-values where relative extrema might occur.
critical number
critical point

31. If 𝑓(𝑐) is defined, and either 𝑓 ′ 𝑐 =0 or 𝑓′(𝑐) DNE, then 𝑥=𝑐 is called a ________________. Also, 𝑐, 𝑓 𝑐 is called a __________________. Critical numbers give us 𝑥-values where relative extrema might occur.
critical number
critical point

32. + − First Derivative Test for Relative Extrema
Let 𝑐 be a critical number. − + 𝑐 𝑥
𝑓′(𝑥)
Relative
max at
𝑐, 𝑓 𝑐

33. − + + − First Derivative Test for Relative Extrema
Let 𝑐 be a critical number. − + 𝑐 𝑥
𝑓′(𝑥)
Relative
max at
𝑐, 𝑓 𝑐 + − 𝑐 𝑥
𝑓′(𝑥)
Relative
min at
𝑐, 𝑓 𝑐

34. + First Derivative Test for Relative Extrema
Let 𝑐 be a critical number. + 𝑐 𝑥
𝑓′(𝑥)
Not a relative
extremum

35. + − First Derivative Test for Relative Extrema
Let 𝑐 be a critical number. + 𝑐 𝑥
𝑓′(𝑥)
Not a relative
extremum − 𝑐 𝑥
𝑓′(𝑥)
Not a relative
extremum

36. Ex 2. Find all critical numbers for 𝑓 𝑥 =2 𝑥 4 −4 𝑥 2 +3, and classify each critical point as a relative maximum, relative minimum, or neither.

37. Ex 2. Find all critical numbers for 𝑓 𝑥 =2 𝑥 4 −4 𝑥 2 +3, and classify each critical point as a relative maximum, relative minimum, or neither.

38. Ex 3. Find all critical numbers for 𝑓 𝑥 = 5−4𝑥− 𝑥 2 , and classify each critical point as a relative maximum, relative minimum, or neither.

39. Ex 3. Find all critical numbers for 𝑓 𝑥 = 5−4𝑥− 𝑥 2 , and classify each critical point as a relative maximum, relative minimum, or neither.