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 $x = 1$
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 $x = 1$