$A$ function $f(x) = \begin{cases} 1 + x, & x \le 2 \\ 5 - x, & x > 2 \end{cases}$ is

Let $k$ and $m$ be positive real numbers such that the function $f(x) = \begin{cases} 3x^2 + k\sqrt{x+1}, & 0 < x < 1 \\ mx^2 + k^2, & x \geq 1 \end{cases}$ is differentiable for all $x > 0$. Then $\frac{8f'(8)}{f'(\frac{1}{8})}$ is equal to $.............$.

Match the items given in List-$I$ with those of the items of List-$II$:

List-$I$	List-$II$
$a$. If $y=|x|+|x-2|$,then at $x=2, \frac{dy}{dx}=$	$i$. $2$
$b$. If $f(x)=|\cos 2x|$,then $f^{\prime}(\frac{\pi}{4}+)=$	$ii$. $0$
$c$. If $f(x)=\sin(\pi[x])$,where $[\cdot]$ denotes the greatest integer function,then $f^{\prime}(1-)=$	$iii$. $-2$
$d$. If $f(x)=\log|x-1|, x \neq 1$,then $f^{\prime}(\frac{1}{2})=$	$iv$. Does not exist

$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}$ equals

The number of real roots of the equation $\log_{e} x + ex = 0$ is

The function $f(x) = |x|$ at $x = 0$ is