The domain of function f (x) = - AX2 + 4x + 1 is [- 1, 2]; (1) if a = 2, find the range of function f (x); (2) if a is a non negative constant and function f (x) is a monotone function on [- 1, 2], find the range of a and the range of function f (x)

The domain of function f (x) = - AX2 + 4x + 1 is [- 1, 2]; (1) if a = 2, find the range of function f (x); (2) if a is a non negative constant and function f (x) is a monotone function on [- 1, 2], find the range of a and the range of function f (x)

(1) When a = 2, f (x) = - 2x2 + 4x + 1 = - 2 (x-1) 2 + 3 & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp;, when a = 2 (2) when x ∈ [- 1,1], f (x) increases monotonically, when x ∈ [1,2], f (x) decreases monotonically, f (x) max = f (1) = 3, and ∵ f (- 1) = - 5, f (2) = 1, ∵ f (x) min = f (- 1) = - 5, ∵ f (x) has a range of [- 5,3] (6 points) (2) when a = 0, f (x) = 4x + 1, monotonically increasing in [- 1, 2] When a > 0, f (x) = − a (x − 2a) 2 + 1 + 4a (8) f (x) & nbsp; In [- 1, 2], the function is monotone ∧ 2A ≤ − 1 or 2A ≥ 2 ∧ 2 ≤ a < 0 or 0 ∧ a ≤ 1 ∧ a > 0 ∧ 0 ∧ a ≤ 1, then the function is monotone increasing in [- 1, 2]. To sum up, when 0 ≤ a ≤ 1, f (x) is monotone increasing in [- 1, 2], f (x) min = f (- 1) = - A-3, f (x) max = f (2) = - 4A + 9, and the value range of a is [- A-3, - 4A + 9], so the value range of a is [0, 1], and the value range of F (x) is [- A-3, - 4A + 9]-----（ 12 points)