Question

Find the intervals in which the function $f$ given by $f\left(x\right)=x^3+\frac{1}{x^3}$, $x\ne 0$ is
(i) increasing (ii) decreasing.

Answer 1

A function $f\left(x\right)$ is increasing if $f^{\prime }\left(x\right)>0$ and decreasing if $f^{\prime }\left(x\right)<0$
$f\left(x\right)=x^3+\frac{1}{x^3}{f}^{\prime }\left(x\right)=3x^2-\frac{3}{x^4}=\frac{3x^6-3}{x^4}$
Now,
$f^{\prime }\left(x\right)>0x\in (-1,0)and(1,\infty )f^{\prime }\left(x\right)<0x\in (-\infty ,-1)andx\in (0,1)$