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Thread: Identifying Inflection Points

  1. #1
    Oct 2007

    Identifying Inflection Points

    The x coordinates of the points of inflection of the graph of $\displaystyle y= x^5-5x^4=3x+7$ are...

    A 0,
    B 1,
    C 3,
    D 0 and 3,
    E 0 and 1

    $\displaystyle y''=20x^3-60x^2$
    $\displaystyle y''=20x^2(x-3)$

    So, thanks to the graph and knowledge of the definition of inflection point (y''= 0).
    I choose D.

    However, the book says C.
    I'm inclined to believe it is right because I did the sign test for 0 for numbers less than 0 and less than 3, thought I may be incorrect of which function to put the test values into (y'' right?), and got the same sign (negative).

    So, did I do the sign test wrong? Or is the right and I'm confusing myself with these questions.

    What is the answer to the problem?

    Thank you.
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  2. #2
    Senior Member JaneBennet's Avatar
    Dec 2007
    For a point of inflection, $\displaystyle y''=0$ is a necessary condition but not a sufficient one. $\displaystyle y''$ must also change sign before and after the point of inflection. Thus 0 is not a point of inflection because $\displaystyle y''$ does not change sign there; $\displaystyle y''$ is negative both slightly before and slightly after 0.

    Alternatively …

    $\displaystyle y'''=60x^2-120x\ne0$ when $\displaystyle x=3$. Since $\displaystyle y'''$ is an odd-order derivative, $\displaystyle x=3$ is a point of inflection. However $\displaystyle y'''(0)=0$ so we differentiate again.

    $\displaystyle y''''=120x-120\ne0$ when $\displaystyle x=0$. $\displaystyle y''''$ is an even-order derivative, so $\displaystyle x=0$ is not a point of inflection.
    Last edited by JaneBennet; Jan 22nd 2008 at 09:16 PM.
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