JEE Advanced2020MathematicsContinuity and DifferentiabilityActual
Let the function f : R → R be defined by f x = x 3 − x 2 + x − 1 sin x and let g : R → R be an arbitrary function. Let f g : R → R be the product function defined by f g x = f x g x . Then which of the following statements is/are TRUE?
Options
- AIf g is continuous at x = 1 , then f g is differentiable at x = 1
- BIf f g is differentiable at x = 1 , then g is continuous at x = 1
- CIf g is differentiable at x = 1 , then f g is differentiable at x = 1
- DIf f g is differentiable at x = 1 , then g is differentiable at x = 1
Correct answer
A. If g is continuous at x = 1 , then f g is differentiable at x = 1
Step-by-step solution
Differentiability of f g at x = 1 Left-hand derivative : f g ' 1 - = lim h → 0 f g 1 - h − f g 1 - h = lim h → 0 1 - h 3 - 1 - h 2 - h sin 1 - h g 1 - h − 0 - h = lim h → 0 1 - h 2 + sin 1 - h g 1 - h   . . . . . . . i Right-hand derivative : f g ' 1 + = lim h → 0 f g 1 + h − f g 1 h = lim h → 0 1 + h 3 - 1 + h 2 + h sin 1 + h g 1 + h − 0 h = lim h → 0 1 + h 2 + sin 1 + h g 1 + h   . . . . . . . i i If g is continuous at x = 1 , then lim