Question

The range of $f\left(x\right)=\frac{1}{|\sin x|}+\frac{1}{|\cos x|}$ is

• A. $(2\sqrt{2},\infty )$
• B. $(\sqrt{2},2\sqrt{2})$
• C. $(0,2\sqrt{2})$
• D. $[2\sqrt{2},4]$

Answer 1

The correct option is A $(2\sqrt{2},\infty )$

Given, $f\left(x\right)=\frac{1}{\mid \sin x\mid }$ + $\frac{1}{\mid \cos x\mid }$

If either of $\sin x$ or $\cos x=0$

Then $f\left(x\right)=\infty$

$\therefore$ maximum of $f\left(x\right)=\infty$

$f\left(x\right)$ will be minimum if $\sin x=\cos x\Rightarrow x=\frac{\pi }{4}$

$\therefore \mid \sin x\mid =\mid \cos x\mid =\frac{!}{\sqrt{2}}$

$\therefore$ minimum of $f\left(x\right)=\sqrt{2}+\sqrt{2}=s\sqrt{2}$

$\therefore$ Range of $f\left(x\right)=[2\sqrt{2},\infty )$