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Homework Help: Inflection point confusion

  1. Jan 8, 2009 #1
    1. The problem statement, all variables and given/known data
    [tex] f(x) = x^{4} - 2x^{2} + 3[/tex]

    Find the intervals of concavity and the inflection points.

    2. Relevant equations
    [tex]f''(x) = 4(3x^{2}-1)[/tex]

    3. The attempt at a solution
    [tex]f''(x)[/tex] is zero at [tex]\pm\frac{1}{\sqrt{3}}[/tex]
    I've found the correct intervals of concavity, which are [tex](-\infty, -\frac{1}{\sqrt{3}}) \cup (\frac{1}{\sqrt{3}}, \infty)[/tex]
    I would expect the inflection points to be [tex]\pm\frac{1}{\sqrt{3}}[/tex], which is partly correct, but the answer the book gives is [tex]\pm\frac{1}{\sqrt{3}}, \frac{22}{9}[/tex].
    I can't see how [tex]\frac{22}{9}[/tex] could be an inflection point. It does not equal zero when plugged in to the second derivative.

  2. jcsd
  3. Jan 8, 2009 #2


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    Science Advisor

    22/9 isn't an "inflection point" but then neither are [itex]1/\sqrt{3}[/itex] nor [itex]1/\sqrt{3}[/itex]! They are not points! What your book is saying is that [itex]\left(1/\sqrt{3}, 22/9\right)[/itex] and [itex]\left(-1/\sqrt{3}, 22/9\right)[/itex] are the inflection points. That is, when x is [itex]1/\sqrt{3}[/itex] or [itex]-1/\sqrt{3}[/itex], y is equal to 22/9.
  4. Jan 8, 2009 #3
    Of course! Thanks for your help
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