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    Derivatives and concavity

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    Find the derivative of f(x)= (3-4x^2)/(x^2-x-6) and then determine intervals of increase/decrease,local max and min values and points of concavity and inflection. Also, please tell me that if the derivative of the function is (4x^2+42x+3)/(x^2-x-6) then do I use the denominator or numerator to determine whether the function is increasing or decreasing?

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    https://brainmass.com/math/derivatives/derivatives-and-concavity-27932

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    f(x)= (3-4x^2)/(x^2-x-6)
    f' = (-8x) /(x^2-x-6) - (3-4x^2)(2x-1) / (x^2-x-6)^2
    = [-8x*(x^2-x-6) - (3-4x^2)(2x-1)] / (x^2-x-6)^2
    = [(-8x^3 + 8x^2+48x) + (8x^3-4x^2-6x+3)] / (x^2-x-6)^2
    = (4x^2+42x+3)/(x^2-x-6)^2

    we will study when does f' = 0
    Since (x^2-x-6)^2 >= 0, we only need to care about the numerator.
    Set 4x^2+42x+3 =0
    Then x = [-42 + ...

    Solution Summary

    This shows how to find the derivative and then determine intervals of increase/decrease,local max and min values and points of concavity and inflection.

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