Let f:[0,3] rightarrow R be defined by f(x)=m | vedclass

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(A) Given $f(x) = \min \{x-[x], 1+[x]-x\}$. Let ${x} = x-[x]$. Then $f(x) = \min \{\{x\}, 1-\{x\}\}$.For $x \in [0, 3)$,the function ${x}$ is periodic

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$5$

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(A) Given $f(x) = \min \{x-[x], 1+[x]-x\}$. Let ${x} = x-[x]$. Then $f(x) = \min \{\{x\}, 1-\{x\}\}$.For $x \in [0, 3)$,the function ${x}$ is periodic with period $1$. The graph of $f(x)$ consists of triangular waves between $0$ and $3$.Specifically,$f(x) = \{x\}$ when $0 \le \{x\} \le 1/2$ and $f(x) = 1-\{x\}$ when $1/2 $1$. Continuity: The function $f(x)$ is continuous everywhere on $[0, 3]$ because ${x}$ is continuous except at integers,but the $\min$ function makes it continuous at integers as well. Thus,$P = \emptyset$ and the number of elements in $P$ is $0$.$2$. Differentiability: The function $f(x)$ is not differentiable at points where the two expressions are equal,i.e.,${x} = 1/2$,and at points where ${x}$ is discontinuous,i.e.,$x \in \{1, 2\}$.In the interval $(0, 3)$,${x} = 1/2$ occurs at $x \in \{1/2, 3/2, 5/2\}$.The points where $f$ is not differentiable are $Q = \{1/2, 1, 3/2, 2, 5/2\}$.The number of elements in $Q$ is $5$.Sum of elements in $P$ and $Q = 0 + 5 = 5$.

List-$I$	List-$II$
$A. \frac{d}{dx}\left(\tan^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)\right)$	$(i) \log(x+\sqrt{1+x^2})$
$B. \frac{d}{dx}\left(\frac{3+\|x-1\|}{3x+4}\right)$	$(ii) -\frac{4x}{(1+x^2)^2}$
$C. \sinh^{-1} x$	$(iii) \frac{1}{2}$
$D. \frac{d^2}{dx^2}\left(\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right)$	$(iv) \frac{1}{\sqrt{1+x^2}}$
	$(v) \text{not differentiable at } x=1$

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