Question

The function $f\left(x\right)={\tan }^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ is

• A. increasing in its domain
• B. decreasing in its domain
• C. decreasing in $(-\infty ,o)$ and increasing in $(0,\infty )$
• D. increasing in$(-\infty ,0)$ and decreasing in $(0,\infty )$

Answer 1

The correct option is D increasing in$(-\infty ,0)$ and decreasing in $(0,\infty )$

Given, $f\left(x\right)={\tan }^{-1}\left(\frac{1-x^2}{1+x^2}\right)$
$f^{\prime }\left(x\right)=-\frac{4x}{(1+x^2{)}^2}$
$f^{\prime }\left(x\right)=0\Rightarrow x=0$
For $x<0,f^{\prime }\left(x\right)>0$
Hence, f(x) is increasing in $(-\infty ,0)$.
For $x>0,f^{\prime }\left(x\right)<0$
Hence, f(x) is decreasing in $(0,\infty )$.