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Math Help - a bit of concavity

  1. #1
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    a bit of concavity

    For what values of x is the graph of y = x4 - 9x both increasing and concave up?.....

    how do i figure this out? i graphed the function, and see where it is concave up, but dont know how to tell what the values are...any help?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by mer1988 View Post
    For what values of x is the graph of y = x4 - 9x both increasing and concave up?.....

    how do i figure this out? i graphed the function, and see where it is concave up, but dont know how to tell what the values are...any help?
    a graph is concave up everywhere it's second derivative is positive
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    yes i know that, but how do i find out the x values?
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  4. #4
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    Quote Originally Posted by mer1988 View Post
    For what values of x is the graph of y = x4 - 9x both increasing and concave up?.....

    how do i figure this out? i graphed the function, and see where it is concave up, but dont know how to tell what the values are...any help?
    y = x^4 -9x
    y' = 4x^3 -9
    y'' = 12x^2

    y' determines the slope of the curve. If y' is negative, the graph is going down. If y' is positive, the graph is going up. If y' = 0, the graph is horizontal, neither decreasing nor increasing. So, y' > 0 if graph is increasing.

    y' = 4x^3 -9
    4x^3 -9 > 0
    x^3 > 9/4
    x > cubrt(2.25) -----the graph is increasing.

    y'' determines the concavity of the curve.
    If the y'' is positive, then the concavity of the graph is facing upwards.

    y'' = 12x^2
    Except for x=0, for any real values of x, the y'' is positive.

    Therefore, for the graph of y = x^4 -9x is increasing and concaves up for all values of x, such that x > cubrt(2.25). -----------answer.
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