Question

If $f\left(x\right)=\sin x+\ln |\sec x+\tan x|-2x$ for $xϵ(-\frac{\pi }{2},\frac{\pi }{2})$ then check the monotonicity of $f\left(x\right)$

• A. Increasing
• B. Decreasing
• C. Increasing for some values and decreasing for other values
• D. None of these

Answer 1

The correct option is A Increasing

$f^{\prime }\left(x\right)=\cos x+|\frac{\sec x.\tan x+{\sec }^{2}x}{\sec x+\tan x}|-2$
$=\cos x-2+|\sec x|$
$=\cos x+|\sec x|-2$
$=\cos x+\frac{1}{|\cos x|}-2$

$=(\sqrt{\cos x}-\frac{1}{\sqrt{\cos x}}{)}^2$
$>0$
Hence
$f^{\prime }\left(x\right)>0$
Hence increasing.