Let f: R R be a function defined by f(x) = cases x^3 - 3x^2 + 6x - 2, & x 1 ax^2 - 2x + b, & x > 1 cases If f(x) is a one-one function on R , then the necessary conditions on a and b are

Step-by-step solution

For f(x) to be one-one on R , it must be strictly monotonic. For x 1 , f'(x) = 3x^2 - 6x + 6 = 3(x-1)^2 + 3 > 0 . Thus, f(x) is strictly increasing on (- , 1] . The range of this piece is (- , f(1)] = (- , 2] . For f(x) to be globally one-one, the second piece must also be strictly increasing, and its range must not overlap with (- , 2] . For x > 1 , f'(x) = 2ax - 2 . For f(x) to be strictly increasing on (1, ) , we need 2ax - 2 0 for all x > 1 . This requires a > 0 and the root x = 1 a 1 , which gives a 1 . Since