Let f x = 1 x 2 : x ≥ 1 α x 2 + β : x < 1 . If f x is continuous and differentiable at any point, then

Step-by-step solution

The given function is clearly continuous at all points except possibly at x = ± 1 . For f x to be continuous at x = 1 , we must have l i m x → 1 - f x = l i m x → 1 + f x = f 1 ⇒ l i m x → 1 - α x 2 + β = l i m x → 1 + 1 x 2 ⇒ α + β = 1 … 1 Now, for f x to be differentiable at x = 1 , we must have l i m x → 1 - f x - f 1 x - 1 = l i m x → 1 + f x - f 1 x - 1 ⇒ l i m x → 1 - α x 2 + β - 1 x - 1 = l i m x → 1 + 1 x 2 - 1 x - 1 ∵ α + β = 1 ∴ β - 1 = - α ⇒ l i m x → 1 - α x 2 - α x - 1 = l i m x → 1 + 1 x 2 - 1 x - 1 ⇒