Question

The function $f\left(x\right)=min\{\sin \sqrt{\left[m\right]}x,\left|x\right|\}$ and $f\left(x\right)$ has period of $\pi$ where $[\cdot ]$ denotes greatest integer function , then

• A. $m\in \left\{4\right\}$
• B. $m\in [3,4)$
• C. $m\in [4,5)$
• D. None of these

Answer 1

Answer: (C)

$m\in [4,5)$

Given $f\left(x\right)=min\{\sin \sqrt{\left[m\right]}x,\left|x\right|\}$
$\Rightarrow f\left(x\right)=\sin \sqrt{\left[m\right]x}(\because \sin px\le \left|x\right|\forall p,x)$
Given ,period of $\sin \sqrt{\left[m\right]x}=\pi$
$\therefore \frac{2\pi }{\sqrt{\left[m\right]}}=\pi$
$\Rightarrow \left[m\right]=4$
$\Rightarrow mϵ[4,5)i.e.4\le m<5$