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Math Help - Increasing/Decreasing Intervals

  1. #1
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    Increasing/Decreasing Intervals

    I cannot figure out how to do this if anyone can help it would be appreciated!

    Find the intervals of increase and decrease for the function f(t) =
    8t/(t+9)^2
    Express your answer in interval notation (be sure to enter a comma to separate the end points of
    the intervals).

    Thanks in advance.
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  2. #2
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    <br />
f(t) = \frac{8t}{(t+9)^2}<br />

    The function is increasing when it has a positive derivative and decreasing when it has a negative derivative.

    f'(t) = \frac{(8)(t+9)^2 - 8t(2)(t+9)}{(t+9)^4} = \frac{8(t^2+18t+81 - 16t(t+9)}{(t+9)^4}

    = \frac{8t^2+144t+648-16t^2-144t}{(t+9)^4} = \frac{648-8t^2}{(t+9)^4}

    Now observe that the denominator is always positive. So we only need to consider the sign of the numerator.

    648 - 8t^2 < 0 when 81 < t^2 which happens when t > 9 or t <-9. Elsewhere the function is positive.
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