The function f (x) = (1 / 3) ^ x, X ∈ [- 1,1]; the minimum value of function g (x) = f ^ 2 (x) - 2AF (x) + 3 is h (a) (1) Find H (a). (2) whether there are real numbers m, N, and meet the following conditions: ① m > n > 3; ② when the definition field of H (a) is [n, M], the value range is [n ^ 2, m ^ 2]. If there is, find the value of M, N; if not, explain the reason

The function f (x) = (1 / 3) ^ x, X ∈ [- 1,1]; the minimum value of function g (x) = f ^ 2 (x) - 2AF (x) + 3 is h (a) (1) Find H (a). (2) whether there are real numbers m, N, and meet the following conditions: ① m > n > 3; ② when the definition field of H (a) is [n, M], the value range is [n ^ 2, m ^ 2]. If there is, find the value of M, N; if not, explain the reason

① Let t = f (x), then G (x) = T ^ 2-2at + 3 deformation: G (x) = (T-A) ^ 2 + 3-A ^ 2 because 1 / 3 < T < 3, then: H (a) = {- 2A / 3 + 28 / 9 a < 1 / 3-A ^ 2 + 3 1 / 3 ≤ a ≤ 3-6a + 12 a > 3. Suppose there is such m, N. then: H (a) = - 6A + 12 A because m > n > 3