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Curve sketch problem

  1. Oct 31, 2009 #1
    1. The problem statement, all variables and given/known data

    Sketch graph of f(x)= (x^2)/((x-2)^2). I have retrieved the first derivative, found the critical points, and also have the vertical asymptote. I seem to be having trouble trying to find the inflection points.... I can't seem to find a nicely factored f''(x).

    2. Relevant equations



    3. The attempt at a solution

    so far I have f''(x)= [ (x-2)^3 ] (-8) [ (3x^2) + x + 2 ] / [(x-2)^8]

    I cant factor the 3x^2+x+2 to be able to find where f''(x)=0 and thus revealing the inflection points :(. Help? thankyou...
     
  2. jcsd
  3. Oct 31, 2009 #2

    Mark44

    Staff: Mentor

    For problems like these, it's more efficient to get the derivative in its simplest form before you take the derivative again.

    For your function, I found this for f'(x):
    [tex]f'(x)~=~\frac{2x(x - 2)^2 - 2x^2(x - 2)}{(x - 2)^4}[/tex]
    By finding common factors in the numerator, I was able to simplify it in this way
    [tex]f'(x)~=~\frac{2x(x - 2)(x - 2 - x)}{(x - 2)^4}~=~ \frac{-4x}{(x - 2)^3}[/tex]

    From there, differentiating to get f''(x) is pretty straightforward.
     
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