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Math Help - Interval question..

  1. #1
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    Interval question..

    I don't get this..

    Find all intervals of increasing&decreasing&all relative extrema. Find all intervals where f is concave up or down. Find all points of inflection.

    f(x) = 6/(x^2+3)

    I am lost on this, no idea how to solve it, I took the derivative and I get my x as sqrt(6i).

    How do I get the concept of these problems? Sometimes I'm able to do them and other times I don't get the concept..
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  2. #2
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    f(x) = \frac{6}{x^2+3}

    f'(x) = \frac{-12x}{(x^2+3)^2}

    f'(x) = 0 at x = 0

    f'(x) > 0 for x < 0 ... f(x) is increasing on the interval (-\infty,0)

    f'(x) < 0 for x > 0 ... f(x) is decreasing on the interval (0, \infty)

    f(x) has an absolute maximum of y = 2 at x = 0


    f''(x) = \frac{36(x^2-1)}{(x^2+3)^3}

    f''(x) = 0 at x = 1 and x = -1

    f''(x) > 0 for x < -1 ... f(x) is concave up

    f''(x) < 0 for -1 < x < 1 ... f(x) is concave down

    f''(x) > 0 for x > 1 ... f(x) is concave up


    since f''(x) changes sign at x = 1 and x = -1, there are inflection points at those x-values.
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