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