Question

Let the function $f: R \rightarrow R$ be defined by $f(x)=x^{3}-x^{2}+(x-1) \sin x$ and let $g: R \rightarrow R$ be an arbitrary function. Let $f g: R \rightarrow R$ 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 $x=1$

Answer 1

Answer: (A), (C)

If $g$ is continuous at $x=1$, then $f g$ is differentiable at $x=1$

Answer 2

(A) If $g$ is continuous at $x=1$, then $f g$ is differentiable at $x=1$
(B) If $g$ is differentiable at $x=1$, then $f g$ is differentiable at $x=1$