Question

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

Answer 1

$f(x)=x^3+\frac{1}{x^3}$
∴$f(x)=3x^2-\frac{3}{x^4}=\frac{3x^6-3}{x^4}$
Then, $f'(x)=0$
$⇒3x^6-3=0$
$⇒x^6=1$
$⇒x=±1$
Now, the points x=1 and x=−1 divide the real line into three disjoint intervals i.e.,$(-∞,-1),(-1,1)$ and $(1,∞)$.
In intervals $(-∞,-1)$ and $(1,∞)$ i.e., when x<−1 and x>1, $f'(x)>0$
Thus, when x<−1 and x>1, f is increasing.
In interval (−1,1) i.e., when $−1<x<1,f'(x)<0$
Thus, when −1<x<1, f is decreasing.