Question

The solution of the inequality $\left(\tan ^{-1} x\right)^2-3 \tan ^{-1} x+2 \geq 0$ is -

• A. $(-\infty, \tan 1] \cup[\tan 2, \infty)$
• B. $(-\infty, \tan 1]$
• C. $(-\infty,-\tan 1] \cup[\tan 2, \infty)$
• D. $[\tan 2, \infty)$

Answer 1

Answer: (B)

$\left(\tan ^{-1} x \right)^2-3 \tan ^{-1} x +2 \geq 0 $
$\left(\tan ^{-1} x -1\right)\left(\tan ^{-1} x -2\right) \geq 0$
we know that $\tan ^{-1} x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
so $\tan ^{-1} x \geq 2$ (not possible) or $\tan ^{-1} x \leq 1$
$\Rightarrow x \in(-\infty, \tan 1]$