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onegirl

  • 3 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).

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  1. bfsgd
    • 3 years ago
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    You can differentiate the function and it is increasing if f ' (x)>0.

  2. onegirl
    • 3 years ago
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    huh?

  3. onegirl
    • 3 years ago
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    @electrokid can u help?

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

  5. onegirl
    • 3 years ago
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    ok

  6. onegirl
    • 3 years ago
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    so what do i do next?

  7. bfsgd
    • 3 years ago
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    That's it :)

  8. onegirl
    • 3 years ago
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    ohh so it will be D?

  9. bfsgd
    • 3 years ago
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    No...D includes negative numbers. You need the one thats only x>1.

  10. electrokid
    • 3 years ago
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    |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
    • 3 years ago
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    Ohh so its B

  12. bfsgd
    • 3 years ago
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    Yep :)

  13. electrokid
    • 3 years ago
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    nope

  14. onegirl
    • 3 years ago
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    what do you mean nope?

  15. onegirl
    • 3 years ago
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    so its wrong?

  16. bfsgd
    • 3 years ago
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    Its right....

  17. electrokid
    • 3 years ago
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    yep. B is not the answer

  18. electrokid
    • 3 years ago
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    check the interval (-1,0)

  19. electrokid
    • 3 years ago
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    in math, there are no shortcuts. follow the "yellow brick road"

  20. onegirl
    • 3 years ago
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    ok

  21. electrokid
    • 3 years ago
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    do you see the mistake?

  22. onegirl
    • 3 years ago
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    yea

  23. electrokid
    • 3 years ago
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    so, what is the answer?

  24. onegirl
    • 3 years ago
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    i got C

  25. electrokid
    • 3 years ago
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    follow the steps. did you use the number line that I drew?

  26. onegirl
    • 3 years ago
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    yea hold on

  27. onegirl
    • 3 years ago
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    so the critical points are -1, 0 and 1

  28. onegirl
    • 3 years ago
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    okay

  29. electrokid
    • 3 years ago
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    sorry. yes. you are correct. "3" points, -1, 0 and 1

  30. onegirl
    • 3 years ago
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    lol ok

  31. electrokid
    • 3 years ago
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    |dw:1364919097503:dw|

  32. electrokid
    • 3 years ago
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    interval (-inf, -1) plug in x=-2 and find f'(x)

  33. onegirl
    • 3 years ago
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    plug in -2 into the problem?

  34. electrokid
    • 3 years ago
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    yep

  35. onegirl
    • 3 years ago
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    into (x - 2)^2?

  36. electrokid
    • 3 years ago
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    into the entire thing f'(x)

  37. onegirl
    • 3 years ago
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    ok

  38. onegirl
    • 3 years ago
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    idk i got 16?

  39. onegirl
    • 3 years ago
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    and the derivative i got 0

  40. electrokid
    • 3 years ago
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    \[f'(-2)=8(-2)[(-2)^2-1]=?\]

  41. onegirl
    • 3 years ago
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    ohh i got -48

  42. electrokid
    • 3 years ago
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    good.|dw:1364919435696:dw| now, we check interval (-1,0) find \[f'(-0.5)\]

  43. onegirl
    • 3 years ago
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    okay i get 0

  44. electrokid
    • 3 years ago
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    nope try again.

  45. onegirl
    • 3 years ago
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    you mean find the derivative of -0.5 ?

  46. electrokid
    • 3 years ago
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    derivative "at" x=-0.5

  47. onegirl
    • 3 years ago
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    ohhh ok

  48. electrokid
    • 3 years ago
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    yep. what did you get?

  49. onegirl
    • 3 years ago
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    i'm still getting 0 :/

  50. electrokid
    • 3 years ago
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    \[f'(-0.5)=8(-0.5)[(-0.5)^2-1]=?\]

  51. onegirl
    • 3 years ago
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    ok i get 3

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

  53. onegirl
    • 3 years ago
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    okay

  54. onegirl
    • 3 years ago
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    i get -3

  55. electrokid
    • 3 years ago
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    great |dw:1364919912982:dw| now the final interval (1,+infty) we take x= 2 \[f'(2)=?\]

  56. onegirl
    • 3 years ago
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    ok i get 48

  57. electrokid
    • 3 years ago
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    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
    • 3 years ago
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    so, in what intervals do you see f'(x) positive?

  59. onegirl
    • 3 years ago
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    ok in 0,1 and 1, + infi

  60. electrokid
    • 3 years ago
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    great. which option says that?

  61. onegirl
    • 3 years ago
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    A!

  62. electrokid
    • 3 years ago
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    and that is your answer yay

  63. onegirl
    • 3 years ago
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    thanks :)

  64. electrokid
    • 3 years ago
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    follow the same procedure for the other one,

  65. onegirl
    • 3 years ago
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    ok

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