Let $f(x) = ax^2 - b|x|$,where $a$ and $b$ are constants. Then at $x = 0$,$f(x)$ has

If $y=\frac{(\sqrt{x}+1)(x^2-\sqrt{x})}{x \sqrt{x}+x+\sqrt{x}}+\frac{1}{15}(3 \cos^2 x-5) \cos^3 x$,then $96 y'(\frac{\pi}{6})$ is equal to :

In the usual notation,the value of $\Delta \nabla$ is equal to

Let $f(x) = \begin{cases} \frac{1}{|x|}, & |x| \geqslant 1 \\ ax^2 + b, & |x| < 1 \end{cases}$ be continuous and differentiable everywhere. Then $a$ and $b$ are

Match the functions in Column $I$ with their properties in Column $II$. In the following $[x]$ denotes the greatest integer less than or equal to $x$.

Column $I$	Column $II$
$A$. $x|x|$	$I$. Strictly increasing and continuous in $(-1,1)$
$B$. $\sqrt{|x|}$	$II$. Continuous but not differentiable in $(-1,1)$
$C$. $x+[x]$	$III$. Differentiable in $(-1,1)$
$D$. $|x-1|+|x+1|+|x|$	$IV$. Differentiable in $(-1,0) \cup (0,1)$
	$V$. Strictly increasing and not differentiable in $(-1,1)$

The correct match is

Let $f:\left[-\frac{1}{2}, 2\right] \rightarrow R$ and $g:\left[-\frac{1}{2}, 2\right] \rightarrow R$ be functions defined by $f(x)=\left[x^2-3\right]$ and $g(x)=|x| f(x)+|4 x-7| f(x)$,where $[y]$ denotes the greatest integer less than or equal to $y$ for $y \in R$. Then
$(A)$ $f$ is discontinuous exactly at three points in $\left[-\frac{1}{2}, 2\right]$
$(B)$ $f$ is discontinuous exactly at four points in $\left[-\frac{1}{2}, 2\right]$
$(C)$ $g$ is $NOT$ differentiable exactly at four points in $\left(-\frac{1}{2}, 2\right)$
$(D)$ $g$ is $NOT$ differentiable exactly at five points in $\left(-\frac{1}{2}, 2\right)$