The smallest and the largest values of ${\tan }^{-1}\left(\frac{1-x}{1+x}\right),0\le x\le 1$  are

${\tan }^{-1}\left(\frac{1-x}{1+x}\right)={\tan }^{-1}1-{\tan }^{-1}x=\frac{\pi }{4}-{\tan }^{-1}x$

$0\le x\le 1\Rightarrow 0\le {\tan }^{-1}x\le \frac{\pi }{4}$

$-\frac{\pi }{4}\le -{\tan }^{-1}x\le 0$

$0\le \frac{\pi }{4}-{\tan }^{-1}x\le \frac{\pi }{4}$

$0\le {\tan }^{-1}\left(\frac{1-x}{1+x}\right)\le \frac{\pi }{4}$

Smallest and longest values of ${\tan }^{-1}\left(\frac{1-x}{1+x}\right)$  are $0,\frac{\pi }{4}$