Find the intervals in which the function $f$ given by
$f(x) = {x^3} + \cfrac{1}{{{x^3}}},x \ne 0$ is

(i) increasing

(ii) decreasing.

Refer Practice Questions, Q.No. 46

(ii) For $f(x)$ to be decreasing function of $x$,

$f'(x) < 0 \Rightarrow 3\left( {{x^2} - \cfrac{1}{{{x^4}}}} \right) < 0$
$\Rightarrow {x^2} - \cfrac{1}{{{x^4}}} < 0 \Rightarrow {x^6} - 1 < 0 \Rightarrow ({x^3} - 1)({x^3} + 1) < 0$

Either ${x^3} - 1 > 0$ and ${x^3} + 1 < 0$

$\Rightarrow {x^3} > 1$ and ${x^3} < - 1 \Rightarrow x > 1$ and $x < - 1,$

which is not possible as there is no common value of $x$.
Or ${x^3} - 1 < 0$ and ${x^3} + 1 > 0$

$\Rightarrow {x^3} < 1$ and ${x^3} > - 1 \Rightarrow x < 1$ and $x > - 1 \Rightarrow - 1 < x < 1.$

Hence, $f(x)$ is decreasing in $( - 1,1)$ .