Question

Complete solution set of the inequality $({\sec }^{-1}x-4)({\sec }^{-1}x-1)({\sec }^{-1}x-2)\ge 0,$ is

• A. $[\sec 2,\sec 1]$
• B. $[\sec 1,\sec 2]\cup [\sec 4,\infty )$
• C. $(-\infty ,\sec 2]\cup [\sec 1,\infty )$
• D. $(-\infty ,\sec 4]\cup [\sec 2,\infty )$

Answer 1

The correct option is C $(-\infty ,\sec 2]\cup [\sec 1,\infty )$

$({\sec }^{-1}x-4)$ is negative,
so $({\sec }^{-1}x-1)({\sec }^{-1}x-2)\le 0$
$1\le {\sec }^{-1}x\le 2$

Hence, $x\in (-\infty ,\sec 2]\cup [\sec 1,\infty )$