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Math Help - Find Concavity and POI

  1. #1
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    Find Concavity and POI

    for  y= x^3-9x^2+27x
    from the first derivative y'=3x^2-18x+27
    If found the critical no x = 3
    and attempted the 1st deriv. test (-inf,3)(3,inf)
    and tested with points 1,4 and got two positives does this mean I have to try with the 2nd derivative
    y''=6x-18
    I'm confused with what I do next....
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  2. #2
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    Quote Originally Posted by radioheadfan View Post
    for  y= x^3-9x^2+27x
    from the first derivative y'=3x^2-18x+27
    If found the critical no x = 3
    and attempted the 1st deriv. test (-inf,3)(3,inf)
    and tested with points 1,4 and got two positives does this mean I have to try with the 2nd derivative
    y''=6x-18
    I'm confused with what I do next....
    y = x^3 - 9x^2 + 27x

    y' = 3x^2 - 18x + 27

    y'' = 6x - 18

    critical values for y' ...

    3x^2 - 18x + 27 = 0

    x^2 - 6x + 9 = 0

    (x -3)^2 = 0

    x = 3

    no extrema at x = 3 because y' does not change signs at x = 3

    critical value for y'' ...

    6x - 18 = 0

    x = 3

    y'' changes sign at x = 3 ... the point (3, 27) is an inflection point on the original function.


    now ... graph the original function and confirm the analysis.
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