Question

Let $f:R\to (0,\infty )$ satisfy the relation $f(x+y)=f\left(x\right)f\left(y\right),f\left(x\right)\ne 0$ for any $x$ and $f\left(x\right)$ is differentiable on $R$ such that $f^{\prime }\left(0\right)=\ln 2.$

	List-I		List-II
(A)	If $\left|\begin{array}{ccc}f\left(x\right)& 1& 0\\ 2& f\left(x\right)& 1\\ 3& 1& 2\end{array}\right|=0,$ then $x$ equals	(P)	$0$
(B)	$\underset{x\to 0}{\text{lim}}\frac{f\left(x\right)-1}{x}$ equals	(Q)	$1$
(C)	Number of points of non-differentiability of $g\left(x\right)=\text{min}\left\{f\right(x),5x-2\}$ is	(R)	$2$
(D)	Number of points of non-differentiability of $h\left(x\right)=\text{min}\left\{f\right(x),[x\left]\right\},$ $x\in (-1,3),$ where $[.]$ denotes the greatest integer function is	(S)	$3$
		(T)	$\ln 2$

Which of the following is CORRECT combination ?

Answer 1

(C) Number of points of non-differentiability is $2$.


$C\to R$

(D) Number of points of discontinuity non-differentiability is $3$
$\because h\left(x\right)=\text{min}\left\{f\right(x),[x\left]\right\}=\left[x\right]$
And $\left[x\right]$ is non-differentiable at integer values.
$\Rightarrow h\left(x\right)$ is non-differentiable at $x=0,1,2$



$D\to S$