Let $f, g: R \rightarrow R$ be two real-valued functions defined as $f(x)=\begin{cases} -|x+3| & , x < 0 \\ e^{x} & , x \geq 0 \end{cases}$ and $g(x)=\begin{cases} x^{2}+k_{1} x & , x < 0 \\ 4 x+k_{2} & , x \geq 0 \end{cases}$,where $k_{1}$ and $k_{2}$ are real constants. If $(g \circ f)$ is differentiable at $x=0$,then $(g \circ f)(-4)+(g \circ f)(4)$ 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

Let $y(x) = (1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$. Then $y'(x) - y''(x)$ at $x = -1$ is equal to:

In which of the following graphs is $x = c$ the point of inflection?

Which of the following is not true?

Let $f$ be a real-valued function defined on the interval $(0, \infty)$ by $f(x)=\ln x+\int_0^x \sqrt{1+\sin t} \, dt$. Then which of the following statement$(s)$ is (are) true?
$(A)$ $f^{\prime \prime}(x)$ exists for all $x \in(0, \infty)$
$(B)$ $f^{\prime}(x)$ exists for all $x \in(0, \infty)$ and $f^{\prime}$ is continuous on $(0, \infty)$,but not differentiable on $(0, \infty)$
$(C)$ there exists $\alpha>1$ such that $|f^{\prime}(x)|<|f(x)|$ for all $x \in(\alpha, \infty)$
$(D)$ there exists $\beta>0$ such that $|f(x)|+|f^{\prime}(x)| \leq \beta$ for all $x \in(0, \infty)$