Let $f(x)$ be a non-negative differentiable function on $[0, \infty)$ such that $f(0)=0$ and $f^{\prime}(x) \leq 2 f(x)$ for all $x>0$. Then,on $[0, \infty)$:

Give the correct order of initials $T$ or $F$ for the following statements. Use $T$ if the statement is true and $F$ if it is false.
Statement-$1$: If $f: R \rightarrow R$ and $c \in R$ is such that $f$ is increasing in $(c - \delta, c)$ and $f$ is decreasing in $(c, c + \delta)$,then $f$ has a local maximum at $c$. Where $\delta$ is a sufficiently small positive quantity.
Statement-$2$: Let $f: (a, b) \rightarrow R, c \in (a, b)$. Then $f$ cannot have both a local maximum and a point of inflection at $x = c$.
Statement-$3$: The function $f(x) = x^2 |x|$ is twice differentiable at $x = 0$.
Statement-$4$: Let $f: [c - 1, c + 1] \rightarrow [a, b]$ be a bijective map such that $f$ is differentiable at $c$ and $f'(c) \neq 0$,then $f^{-1}$ is also differentiable at $f(c)$.

Match the items of List-$I$ with those 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 $f'(\frac{\pi}{4} +) =$	$II$. $0$
$C$. If $f(x) = \sin(\pi[x])$, where $[x]$ is the greatest integer function, then $f'(1-) =$	$III$. $-2$
$D$. If $f(x) = \log|x - 1|$, $x \neq 1$, then $f'(\frac{1}{2}) =$	$IV$. does not exist

Let $g(x)$ be a linear function and $f(x) = \begin{cases} g(x) & , x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} & , x > 0 \end{cases}$ is continuous at $x = 0$. If $f^{\prime}(1) = f(-1)$,then the value of $g(3)$ is

Let $R$ be the set of all real numbers and $f(x) = \sin^{10} x (\cos^8 x + \cos^4 x + \cos^2 x + 1)$ for $x \in R$. Let $S = \{\lambda \in R : \text{there exists a point } c \in (0, 2\pi) \text{ with } f'(c) = \lambda f(c)\}$. Then,

Let $f:(-2,2) \rightarrow \mathbb{R}$ be defined by $f(x) = \begin{cases} x[x] & , -2 < x < 0 \\ (x-1)[x] & , 0 \leq x < 2 \end{cases}$ where $[x]$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in $(-2,2)$ at which $y = |f(x)|$ is not continuous and not differentiable,then $m + n$ is equal to $...........$.