## onegirl 2 years ago For what intervals is f(x) = 2x^4 – 4x^2 + 6 increasing? A. The graph is increasing on the intervals (-1, 0) and (1, ∞). B. The graph is increasing on the interval (1, ∞). C. The graph is increasing on the interval (-1, 0). D. The graph is increasing on the intervals ( -∞ , -1) and (0, 1).

1. bfsgd

You can differentiate the function and it is increasing if f ' (x)>0.

2. onegirl

huh?

3. onegirl

@electrokid can u help?

4. bfsgd

Differentiating f(x) we get \[f'(x)=8x^3 - 8x=8x(x^2-1)\] That is positive for all x>1.

5. onegirl

ok

6. onegirl

so what do i do next?

7. bfsgd

That's it :)

8. onegirl

ohh so it will be D?

9. bfsgd

No...D includes negative numbers. You need the one thats only x>1.

10. electrokid

|dw:1364917996121:dw| get the critical points you see the four intervals? take an arbitrary number from each interval and check the signs of the derivative

11. onegirl

Ohh so its B

12. bfsgd

Yep :)

13. electrokid

nope

14. onegirl

what do you mean nope?

15. onegirl

so its wrong?

16. bfsgd

Its right....

17. electrokid

yep. B is not the answer

18. electrokid

check the interval (-1,0)

19. electrokid

20. onegirl

ok

21. electrokid

do you see the mistake?

22. onegirl

yea

23. electrokid

24. onegirl

i got C

25. electrokid

follow the steps. did you use the number line that I drew?

26. onegirl

yea hold on

27. onegirl

so the critical points are -1, 0 and 1

28. onegirl

okay

29. electrokid

sorry. yes. you are correct. "3" points, -1, 0 and 1

30. onegirl

lol ok

31. electrokid

|dw:1364919097503:dw|

32. electrokid

interval (-inf, -1) plug in x=-2 and find f'(x)

33. onegirl

plug in -2 into the problem?

34. electrokid

yep

35. onegirl

into (x - 2)^2?

36. electrokid

into the entire thing f'(x)

37. onegirl

ok

38. onegirl

idk i got 16?

39. onegirl

and the derivative i got 0

40. electrokid

\[f'(-2)=8(-2)[(-2)^2-1]=?\]

41. onegirl

ohh i got -48

42. electrokid

good.|dw:1364919435696:dw| now, we check interval (-1,0) find \[f'(-0.5)\]

43. onegirl

okay i get 0

44. electrokid

nope try again.

45. onegirl

you mean find the derivative of -0.5 ?

46. electrokid

derivative "at" x=-0.5

47. onegirl

ohhh ok

48. electrokid

yep. what did you get?

49. onegirl

i'm still getting 0 :/

50. electrokid

\[f'(-0.5)=8(-0.5)[(-0.5)^2-1]=?\]

51. onegirl

ok i get 3

52. electrokid

good.|dw:1364919843444:dw| now we check interval (0,1) lets take x = 0.5 similarly, find \[f'(0.5)\]

53. onegirl

okay

54. onegirl

i get -3

55. electrokid

great |dw:1364919912982:dw| now the final interval (1,+infty) we take x= 2 \[f'(2)=?\]

56. onegirl

ok i get 48

57. electrokid

perfect. |dw:1364919991818:dw| now, f(x) is increasing when f'(x) >0 (i.e., f'(x) is positive) and f(x) is decreasing when f'(x) < 0 (i.e., f'(x) is negative)

58. electrokid

so, in what intervals do you see f'(x) positive?

59. onegirl

ok in 0,1 and 1, + infi

60. electrokid

great. which option says that?

61. onegirl

A!

62. electrokid

63. onegirl

thanks :)

64. electrokid

follow the same procedure for the other one,

65. onegirl

ok