Consider a function g x = f x - 2 , ∀ x ∈ R , where f x = 1 x : x ≥ 1 a x 2 + b : x < 1 . If g x is continuous as well as differentiable for all x , then

Step-by-step solution

For g x to be continuous and differentiable ∀ x ∈ R , f x must be continuous and differentiable ∀ x ∈ R Since, f ( x ) is continuous for all x , therefore, it is continuous at x = 1 also. ∴ f 1 = l i m h → 0 + f 1 - h = 1 = l i m h → 0 + a 1 - h 2 + b ⇒ a + b = 1 … 1 Also, f x is differentiable at x = 1 ⇒ f ' 1 − = f ' 1 + ⇒ l i m h → 0 + f 1 - h - f 1 - h = l i m h → 0 + f 1 + h - f 1 h ⇒ l i m h → 0 + a 1 - h 2 + b - 1 - h = l i m h → 0 + 1 1 + h - 1 h ⇒ l i m h → 0 + a + b - 1 + h 2 - 2 h a - h = l i m h → 0 + 1