Question

Let the function f : R → R be defined by f(x) = x³ – x² + (x – 1) sin x 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 ? 

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

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

(C) If g is differentiable at x = 1, then fg is differentiable at x = 1 

(D) If fg is differentiable at x = 1, then g is differentiable at x = 1 Ans. (A,C)

Answer 1

(A) If g is continuous at x = 1, then fg is differentiable at x = 1 (C) If g is differentiable at x = 1, then fg is differentiable at x = 1 

If g(x) is differentiable then h(x) is also differentiable (true) option (A). If g(x) is continuous at x = 1 then g(1+ ) = g(1– ) = g(1)

Option (B) (D)

So we cannot comment on the continuity and differentiability of the function.