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Points of Inflection on a rational function

  1. Aug 5, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the inflection points and use second derivative test to determine where the function is concave up or down

    2. Relevant equations

    f(x) = (x - 1)/(x2 - 4)

    3. The attempt at a solution

    f'(x) = (-x2 + 2x - 4) / (x2 - 4)2

    f''(x) = 2x3 - 6x2 +24x -8 / (x2-4)4

    This is where I am stuck. I can't solve the numerator set to 0. You can factor out the 2 and thats about it. Cubic that I cant solve. I'm looking at:

    x(x2 - 3x +12) = 4

    and thinking that maybe I can use the quadratic equation to find when that quadratic = 0 but I dont think that will help me.
     
  2. jcsd
  3. Aug 5, 2012 #2

    HallsofIvy

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    You want to solve [itex]2x^3- 6x^2+ 24x- 8= 0[/itex]? The obvious first thing to do is to divide through by 2 to get [itex]x^3- 3x^2+ 12x- 4= 0[/itex]. Now the only possible rational roots are [itex]\pm 1, \pm 2[/itex], and [itex]\pm 4[/itex]. Trying those in turn, we see that none of them satisfy the equation so there is not any "simple" solution.
     
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