Let the function f : ℝ → ℝ be defined by f ( x ) = x 3 − x 2 + ( x − 1)sin x and let g …

Mathematics · JEE Advanced · NTA Exams — Limit, Continuity and Differentiability

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
(fg)(x) = f(x)g(x). Then which of the following statements is/are TRUE?
  1. If g is continuous at x = 1, then fg is differentiable at x = 1 .
  2. If fg is differentiable at x = 1, then g is continuous at x = 1 .
  3. If g is differentiable at x = 1, then fg is differentiable at x = 1 .
  4. If fg is differentiable at x = 1, then g is differentiable at x = 1

Answer

(B) If fg is differentiable at x = 1, then g is continuous at x = 1 .

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