Let $f (x) = \frac{(x - 3) (x + 2) (x + 6)}{(x + 1) (x - 5)}$
Find intervals, where $f (x)$ is positive or negative.

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Solution

Let $f (x) = \frac{(x - 3) (x + 2) (x + 6)}{(x + 1) (x - 5)}$
The critical points are $- 6, - 2, - 1, 3, 5$
For $f (x) > 0,$ for all $x \in (- 6, - 2) \cup (- 1, 3) \cup (5, \infty)$
For $f (x) < 0,$ for all $x \in (- \infty, - 6) \cup (- 2, - 1) \cup (3, 5)$

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