Question

For the function $f\left(x\right)=\text{cosec}x-\sec x$, which of the following is true regarding its monotonicity at $x=\frac{\pi }{4}$

• A. Increasing
• B. Decreasing
• C. Non-monotonic
• D. Cannot be determined

Answer 1

The correct option is B Decreasing

$f\left(x\right)=\text{cosec}x-\sec x$
$\Rightarrow f^{\prime }\left(x\right)=-\text{cosec}x\cdot \cot x-\sec x\cdot \tan x$
$\Rightarrow f^{\prime }\left(\frac{\pi }{4}\right)=-\sqrt{2}-\sqrt{2}=-2\sqrt{2}<0$
$\Rightarrow f\left(x\right)$ is decreasing.