Let the function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by $f ( x ) = x ^ { 3 } - x ^ { 2 } + ( x - 1 ) \sin x$ and let $g : \mathbb { R } \rightarrow \mathbb { R }$ be an arbitrary function. Let $f g : \mathbb { R } \rightarrow \mathbb { R }$ be the product function defined by $( f g ) ( x ) = f ( x ) g ( x )$. Then which of the following statements is/are TRUE?
(A) If $g$ is continuous at $x = 1$, then $f g$ is differentiable at $x = 1$
(B) If $f g$ is differentiable at $x = 1$, then $g$ is continuous at $x = 1$
(C) If $g$ is differentiable at $x = 1$, then $f g$ is differentiable at $x = 1$
(D) If $f g$ is differentiable at $x = 1$, then $g$ is differentiable at $x = 1$

If the function $g(x) = \begin{cases} k\sqrt{x+1}, & 0 \leq x \leq 3 \\ mx + 2, & 3 < x \leq 5 \end{cases}$ is differentiable, then the value of $k + m$ is:
(1) $2$
(2) $\frac{16}{5}$
(3) $\frac{10}{3}$
(4) $4$

The function $f ( x ) = \left| x ^ { 2 } - 2 x - 3 \right| \cdot \mathrm { e } ^ { 9 x ^ { 2 } - 12 x + 4 }$ is not differentiable at exactly :
(1) Four points
(2) Two points
(3) three points
(4) one point

$f , g : R \rightarrow R$ be two real valued function defined as $f ( x ) = \left\{ \begin{array} { c l } - | x + 3 | & , x < 0 \\ e ^ { x } & , x \geq 0 \end{array} \right.$ and $g ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } + k _ { 1 } x , & x < 0 \\ 4 x + k _ { 2 } , & x \geq 0 \end{array} \right.$, where $k _ { 1 }$ and $k _ { 2 }$ are real constants. If $gof$ is differentiable at $x = 0$, then $gof ( - 4 ) + gof ( 4 )$ is equal to
(1) $4 \left( e ^ { 4 } + 1 \right)$
(2) $2 \left( 2 e ^ { 4 } + 1 \right)$
(3) $4 e ^ { 4 }$
(4) $2 \left( 2 e ^ { 4 } - 1 \right)$

Let $F ( x )$ be a polynomial with real coefficients and $F ^ { \prime } ( x ) = f ( x )$ . It is known that $f ^ { \prime } ( x ) > x ^ { 2 } + 1.1$ holds for all real numbers $x$. Select the correct options.
(1) $f ^ { \prime } ( x )$ is an increasing function
(2) $f ( x )$ is an increasing function
(3) $F ( x )$ is an increasing function
(4) $[ f ( x ) ] ^ { 2 }$ is an increasing function
(5) $f ( f ( x ) )$ is an increasing function