Question

Let $f\left(x\right)=\{\begin{array}{cc}\frac{\sin (x-[x\left]\right)}{x-\left[x\right]},& x\in (-2,-1)\\ max\{2x,3[|x|\left]\right\},& |x|<1\\ 1& \text{otherwise}\end{array},$ where $\left[t\right]$ denotes greatest integer $\le t$. If $m$ is the number of points where $f$ is not continuous and $n$ is the number of points where $f$ is not differentiable, then the ordered pair $(m,n)$ is

• A. $(3,3)$
• B. $(2,3)$
• C. $(2,4)$
• D. $(3,4)$

Answer 1

The correct option is B $(2,3)$

Given, $f\left(x\right)=\{\begin{array}{cc}\frac{\sin (x-[x\left]\right)}{x-\left[x\right]},& x\in (-2,-1)\\ max\{2x,3[|x|\left]\right\},& |x|<1\\ 1& \text{otherwise}\end{array}$

$f\left(x\right)=\{\begin{array}{ccc}\frac{\sin (x+2)}{x+2},& x\in (-2,-1)& \\ 0,& -1<x\le 0& \\ 2x,& 0\le x<1& \\ 1,& \text{otherwise}& \end{array}$

It clearly shows that $f\left(x\right)$ is discontinuous at $x=-1,1$ also non differentiable.

and at $x=0$,
$L.H.D.=\underset{h\to 0}{lim}\frac{f(0-h)-f\left(0\right)}{-h}=0$

$R.H.D=\underset{h\to 0}{lim}\frac{f(0+h)-f\left(0\right)}{h}=2$
$\therefore f\left(x\right)$ is not differentiable at $x=0.$

Hence, the value of $m=2$ and $n=3.$