Question

Function $f\left(x\right)=\log \left(1+x\right)-\frac{2x}{1+x}$ is monotonic increasing when

• A. $-1<x<0$
• B. $x>1$
• C. $x\in R$
• D. $x\in \left(0,1\right)$

Answer 1

The correct option is B $x>1$

$f\left(x\right)=\log (1+x)=\frac{2x}{1+x}$

$f;\left(x\right)=\frac{1}{1+x}=\frac{2(1+x)-2x}{(1+x{)}^2}$

$=\frac{1}{1+x}-\frac{2}{(1+x{)}^2}=\frac{1+x-2}{(1+x{)}^2}=\frac{x-1}{(x+1{)}^2}$

For monotonic increasing

$f^{\prime }\left(x\right)>0$

$\Rightarrow \frac{x-1}{(x+1{)}^2}>0$

$\Rightarrow x-1>0$

$\Rightarrow x>1$

$f\left(x\right)=\log (1+x)-\frac{2x}{1+x}cs$ monotonic increasing

when $x>1$