Question

The function $f\left(x\right)=x+\ln \left(\frac{x}{1+x}\right)$ is increasing in

• A. $(-1,0)$
• B. $(-\infty ,\infty )$
• C. only some part of its domain
• D. its domain

Answer 1

The correct option is D its domain

Given : $f\left(x\right)=x+\ln \left(\frac{x}{1+x}\right)$
For domain of $f\left(x\right):\frac{x}{1+x}>0$
$\Rightarrow x\in (-\infty ,-1)\cup (0,\infty )$

Now, $f^{\prime }\left(x\right)=1+\frac{1+x}{x}\times \left(\frac{1}{(1+x{)}^2}\right)$
$=\frac{x(1+x)+1}{x(1+x)}=\frac{x^2+x+1}{x(1+x)}$
For increasing, $f^{\prime }\left(x\right)\ge 0$
$\Rightarrow x(1+x)>0\{\because x^2+x+1>0\text{as}a>0,D<0\}$

$\Rightarrow x\in (-\infty ,-1)\cup (0,\infty )$ which is the domain of the function.