Browse Questions

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

Toolbox:
• $a^2-b^2=(a+b)(a-b)$
• $\large\frac{d}{dx}$$(x^n)=nx^{n-1} Step 1: We have f(x)=x^3+\large\frac{1}{x^3},$$x\neq 0$
$f'(x)=3x^2-\large\frac{3}{x^4}$
For $f(x)$ is an increasing function of $x$,
$f'(x)>0$
(i.e) $3\big(x^2-\large\frac{1}{x^4}$$\big)>0 \Rightarrow x^6-1>0\Rightarrow (x^3-1)(x^3+1)>0 x^3-1>0 and x^3+1>0 \Rightarrow x^3>1 & x^3>-1 \Rightarrow x>1 & x>-1 \Rightarrow x^3-1>0 and x^3+1<0 x^3<1 and x^3<-1 x<1 and x<-1 \Rightarrow x<-1 Hence f(x) is increasing when x<-1 and x>1 Step 2: For f(x) to be decreasing function of x f'(x)<0 i.e 3\big(x^2-\large\frac{1}{x^4}\big)$$<0$