Let the function f: {R} \to{R} be defined by f(x)=x^{3}-x^{2}+(x-1) \sin x and let g: {R} \to {R} be an arbitrary function. Let f g: {R} \to R be the product function defined by (f, g)(x)=f(x) g(x) . Then which of the following statements is/are TRUE? [JEE Adv 2020 P1]

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

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

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

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

Limits, Continuity and Differentiability

Section: Mathematics

Question : 18 of 51

Marks: +1, -0

Let the function f:R→R be defined by f(x)=x3−x2+(x−1)‌sin‌x and let g:R→R be an arbitrary function. Let fg:R→R be the product function defined by (f,g)(x)=f(x)g(x). Then which of the following statements is/are TRUE?

[JEE Adv 2020 P1]

If g is continuous at x=1, then fg is differentiable at x=1
If fg is differentiable at x=1, then g is continuous at x=1
If g is differentiable at x=1, then fg is differentiable at x=1
If fg is differentiable at x=1, then g is differentiable at x=1

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