We have, $f(x)=x|\sin x|, x \in R$
$f(x)=\left\{\begin{array}{ll} x \sin x, & x \in(2 n \pi,(2 n+1) \pi) \\ -x \sin x, & x \in((2 n+1) \pi, 2 n \pi) \end{array}\right.$
$f^{\prime}(n \pi)=\lim _{x \rightarrow n \pi} \frac{f(x)-f(n \pi)}{x-n \pi}$
$=\lim _{x \rightarrow n \pi} \frac{x \sin x \mid}{x-n \pi}$
Clearly, $f(x)$ is differentiable for all $x$ except $x=n \pi, n=\pm 1, \pm 2, \pm 3, \ldots$
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| Column $I$ | Column $II$ |
| $(A)$ The minimum value of $\frac{x^2+2 x+4}{x+2}$ is | $(p)$ $0$ |
| $(B)$ Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skewsymmetric, and $(A+B)(A-B)=(A-B)(A+B)$. If $(A B)^t=(-1)^k A B$, where $(A B)^t$ is the transpose of the matrix $A B$, then the possible values of $k$ are | $(q)$ $1$ |
| $(C)$ Let $\mathrm{a}=\log _3 \log _3 2$. An integer $\mathrm{k}$ satisfying $1<2^{\left(-k+3^{-2}\right)}<2$, must be less than | $(r)$ $2$ |
| $(D)$ If $\sin \theta=\cos \phi$, then the possible values of $\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)$ are | $(s)$ $3$ |
$\frac{\text{a}+\text{b}}{2}\int\limits^\text{b}_\text{a}\text{f}(\text{b}-\text{x})\text{dx}$
$\frac{\text{a}+\text{b}}{2}\int\limits^\text{b}_\text{a}\text{f}(\text{b}+\text{x})\text{dx}$
$\frac{\text{b}-\text{a}}{2}\int\limits^\text{b}_\text{a}\text{f}(\text{x})\text{dx}$
$\frac{\text{a}+\text{b}}{2}\int\limits^\text{b}_\text{a}\text{f}(\text{x})\text{dx}$