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.