Math Help - Increasing/Decreasing Intervals

1. 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).

2. $
f(t) = \frac{8t}{(t+9)^2}
$

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.