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  1. #1
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    concave

    For what values of x is the graph of y=xe^−4x concave down?
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  2. #2
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    Quote Originally Posted by cwarzecha View Post
    For what values of x is the graph of y=xe^−4x concave down?
    The x values for which the second derivative is positive.
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  3. #3
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    ok sorry i should've been more specific. i don't know how to find the second derivative of that.
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  4. #4
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    Quote Originally Posted by cwarzecha View Post
    For what values of x is the graph of y=xe^?4x concave down?
    Use second derivative test to see where the function is concave or convex.
    Last edited by 11rdc11; October 14th 2008 at 09:26 PM. Reason: man im slow to responding today lol
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    whats the second derivative test?
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  6. #6
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    Quote Originally Posted by cwarzecha View Post
    For what values of x is the graph of y=xe^−4x concave down?
    We have a product of functions, so we use the product rule.

    \frac{dy}{dx} = e^{-4x} - 4xe^{-4x} = e^{-4x}(1 - 4x).

    \frac{d^2y}{dx^2} = -4e^{-4x} - 4e^{-4x}(1 - 4x) = -4e^{-4x}(2 - 4x).

    We require that this be greater than 0.

    Notice that -4e^{-4x} is always negative.

    So for the second derivative to be positive, 2 - 4x must also be negative (as negative times negative is positive).

    2 - 4x < 0

    2 < 4x

    x > \frac{1}{2}.


    So the values of x for which the graph is concave down are all those that are greater than \frac{1}{2}.
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