Match the following:In the following,[x] deno | vedclass

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(C) $f(x) = x|x| = \begin{cases} x^2, & x \geq 0 \ -x^2, & x < 0 \end{cases}$. The derivative $f'(x) = 2|x|$ exists for all $x$,so it is differentiab

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$a-(ii), b-(iv), c-(iii), d-(i)$

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(C) $f(x) = x|x| = \begin{cases} x^2, & x \geq 0 \ -x^2, & x $(b)$ $f(x) = \sqrt{|x|}$. At $x=0$,the left-hand derivative is $\lim_{h 	o 0^-} \frac{\sqrt{-h}-0}{h} = -\infty$ and right-hand derivative is $\infty$. Thus,it is not differentiable at $x=0$,which is in $(-1, 1)$. Thus,$(b) 	o (iv)$.$(c)$ $f(x) = x+[x]$. This function is strictly increasing in $(-1, 1)$ because $x$ is strictly increasing and $[x]$ is non-decreasing. Thus,$(c) 	o (iii)$.$(d)$ $f(x) = |x-1|+|x+1|$. In $(-1, 1)$,$f(x) = -(x-1) + (x+1) = 2$,which is a constant function and therefore continuous in $(-1, 1)$. Thus,$(d) 	o (i)$.The correct matching is $a-(ii), b-(iv), c-(iii), d-(i)$.

$(a)$ $x\|x\|$	$(i)$ continuous in $(-1, 1)$
$(b)$ $\sqrt{\|x\|}$	$(ii)$ differentiable in $(-1, 1)$
$(c)$ $x+[x]$	$(iii)$ strictly increasing in $(-1, 1)$
$(d)$ $\|x-1\|+\|x+1\|$	$(iv)$ not differentiable at,at least one point in $(-1, 1)$

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