If α and β are such that the function f ( x ) defined by f x = α x 2 - β , for | x | < 1 - 1 | x | , for | x | ≥ 1 is differentiable everywhere then the ordered pair ( α , β ) =

Step-by-step solution

It is given that the function is differentiable at all points, By the first principle, LHD at   x = 1 = RHD at   x = 1 lim x → 1 - f ( x ) - f ( 1 ) x - 1 = lim x → 1 + f ( x ) - f ( 1 ) x - 1 lim x → 1 - α x 2 - β - α + β x - 1 = lim x → 1 + - 1 x + 1 x - 1 lim x → 1 - α x 2 - 1 x - 1 = lim x → 1 + x - 1 x x - 1 lim x → 1 - α ( x + 1 ) ( x - 1 ) x - 1 = lim x → 1 + 1 x lim x → 1 - α x + 1 = lim x → 1 + 1 x &#94