Question

The range of the function $f\left(x\right)=3\mid \sin x\mid -2\mid \cos x\mid$ is -

• A. $[-\sqrt{13},\sqrt{13}]$
• B. $[-2,3]$
• C. $[3,\sqrt{13}]$
• D. $[0,\sqrt{13}]$

Answer 1

The correct option is A $[-\sqrt{13},\sqrt{13}]$

Then $f\left(x\right)=3\sin x-2\cos x$.

Let, $3=r\cos a,2=r\sin a$, where $r=\sqrt{13}$ and $a={\tan }^{-1}\frac{2}{3}$.

Then $f\left(x\right)=r\sin (x-a)$.

As $-1\le \sin (x-a)\le 1$

$\Rightarrow -r\le r\sin (x-a)\le r$

or, $-\sqrt{13}\le r\sin (x-a)\le \sqrt{13}$.

$-\sqrt{13}\le r\sin (x-a)\le \sqrt{13}$.

So, the range of the function $f\left(x\right)$ is $[-\sqrt{13},\sqrt{13}]$.