If the derivative of the function f(x) = cases ax^2 + b & if x < -1 bx^2 + ax + 4 & if x -1 cases is continuous everywhere then

Step-by-step solution

For f(x) to have a continuous derivative everywhere, f(x) must be differentiable and hence continuous at x = -1 . Equating the left-hand and right-hand limits of f(x) at x = -1 : a(-1)^2 + b = b(-1)^2 + a(-1) + 4 a + b = b - a + 4 2a = 4 a = 2 The derivative of the function is: f'(x) = cases 2ax & if x -1 cases Since f'(x) is continuous at x = -1 , its left-hand and right-hand limits at x = -1 must be equal: 2a(-1) = 2b(-1) + a -2a = -2b + a 2b = 3a Substituting a = 2 : 2b = 3(2) b = 3 Answer: a = 2, b = 3