It is known that the function y = f (x) is an odd function defined on R. when x0, the analytic expression of F (x) is obtained (2) Are there real numbers a and B (where 0 ≤ a ＜ b) such that the value range of F (x) on [a, b] is [a, b]? If so, find out all the values of a and B. if not, explain the reason

It is known that the function y = f (x) is an odd function defined on R. when x0, the analytic expression of F (x) is obtained (2) Are there real numbers a and B (where 0 ≤ a ＜ b) such that the value range of F (x) on [a, b] is [a, b]? If so, find out all the values of a and B. if not, explain the reason

F (x) is an odd function, so f (x) + F (- x) = 0
(1) F (- 1) = - 1, so f (1) = - f (- 1) = 1
x> 0, f (x) = - f (- x) = - ((- x) ^ 2 + 2 (- x)) = - x ^ 2 + 2x
(2) When x > 0, f (x) = - x ^ 2 + 2x = - (x-1) ^ 2 + 1