Question

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

Answer 1

We have to find the smallest and largest value of ${\tan }^{-1}\left(\frac{1-x}{1+x}\right)$ for $0\le x\le 1$

We know $0\le x\le 1$

$⟹0\ge -x\ge -1$

$⟹1-1\le 1-x\le 0+1$

$⟹0\le 1-x\le 1$ ------ (i)

Similarly,

$0\le x\le 1$

$⟹1\le 1+x\le 2$ ------ (ii)

Divide(1) by (ii)

$0\le \frac{1-x}{1+x}\le \frac{1}{2}$

$0\le {\tan }^{-1}\frac{1-x}{1+x}\le 0.463radian$

Therefore, smallest value is $0$ and the largest value is $0.463radian$