Find the interval in which the following function is increasing or decreasing.
$f (x) = 3 x^{4} - 4 x^{3} - 12 x^{2} + 5$

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Solution

$f (x) = 3 x^{4} - 4 x^{3} - 12 x^{2} + 5$

$f^{'} (x) = 12 x^{3} - 12 x^{2} - 24 x$
$= 12 x (x^{2} - x - 2)$
$= 12 x (x + 1) (x - 2)$

for $f (x)$ to be increasing $f^{'} (x) > 0$
i.e $12 x (x + 1) (x - 2) > 0$

$f (x)$ is increasing in $(- 1, 0) \cup (2, \infty)$
deceasing in $(- \infty, - 1) \cup (0, 2)$

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