JEE Advanced2020MathematicsContinuity and DifferentiabilityActual
Let f : ℝ → ℝ and g : ℝ → ℝ be functions satisfying f x + y = f x + f y + f x f y and f x = x g x for all x , y ∈ ℝ . If lim x → 0 g x = 1 , then which of the following statements is/are TRUE?
Options
- Af is differentiable at every x ∈ ℝ
- BIf g 0 = 1 , then g is differentiable at every x ∈ ℝ
- CThe derivative f ' 1 is equal to 1
- DThe derivative f ' 0 is equal to 1
Correct answer
A. f is differentiable at every x ∈ ℝ
Step-by-step solution
Given f x + y = f x + f y + f x f y Put x = y = 0 in given relation. ⇒    f 0 = f 0 + f 0 + f 2 0 ⇒    f 0 = 0 or - 1 ∵    f x + y = f x + f y + f x · f y ⇒    f x + y - f x y = f y 1 + f x y ⇒    lim y → 0 f ( x + y ) - f ( x ) y = lim y → 0 1 + f x · f y y ∵    lim x → 0   g x = lim x → 0 f x x = 1 ⇒    f ' x = 1 + f x ⇒    f ' 0 = 1 + f 0 &