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Thread: a question about inflection points

  1. #1
    Junior Member
    Oct 2009

    a question about inflection points

    If there is a maximum and a minimum on a curve, will the inflection point between them be exactly in the middle? Or can it be located anywhere within the distance between the maximum and the minimum?
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  2. #2
    MHF Contributor matheagle's Avatar
    Feb 2009
    I don't think so.
    Let $\displaystyle y=ax^3+bx^2$

    Then $\displaystyle y'=3ax^2+2bx$ giving maxs/mins at x=0,-2b/(3a)

    And $\displaystyle y''=6ax+2b$ giving infl pts at x=-b/(3a)
    which is half way inbetween.

    Now even if we change that to $\displaystyle y=ax^3+bx^2+cx+d$
    the inflection pt is the same (x that is)
    while the max/min is the solution to $\displaystyle 3ax^2+2bx+c=0$
    which gives $\displaystyle x={-2b\pm\sqrt{4b^2-12ac}\over 6a}$ and that average is once again -b/(3a)

    So it is true for polynomials of degree 3.

    Moving onto the next power...

    Try $\displaystyle y'=x^3-3x^2+2x$ giving mins at x=0 and 2 and a max at 1

    while $\displaystyle y''=3x^2-6x+2$ giving infl pts at $\displaystyle x=1\pm 1/\sqrt{3}$

    and those two inflection pts are not at .5 and 1.5, the midpoints of 0 and 1, and of 1 and 2.
    Last edited by matheagle; Nov 20th 2009 at 11:56 PM.
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