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Determining concavity of f(x) = x^2 + 2x/(x-2)^2

  1. May 22, 2008 #1
    Hi, sorry to disturb you,
    But with the equation f(x)= x^2+2x/(x-2)^2

    I need to find the intervals at which f(x) is concave up, and down.

    I found f '(x)= (2x+2)(x-2)^2 -2(x-2)(x^2+x)/((x-2)^4)

    From there I equated it to 0, and found the critical points to be x=2,-2/3.

    However, I have just noticed there is a point of inflection at x=-2. How did I miss this point in my calculation for critical points, any help would be much obliged.

    Last edited: May 22, 2008
  2. jcsd
  3. May 22, 2008 #2


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    A point of inflection is not necessarily a critical point. To know about concavity and points of inflection you need to compute the second derivative.
  4. May 22, 2008 #3


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    To find points of inflection, find the values of x for which f''(x) is zero.
    Last edited: May 22, 2008
  5. May 24, 2008 #4


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    More specifically, where the second derivative changes sign. A point where the second derivative is 0 may not be a point of inflection.
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