It is known that the quadratic function f (x) = ax & # 178; + BX + C (a, B, C belong to R) satisfies the following conditions: (1) when x belongs to R, the minimum value of F (x) is 0 It is known that the quadratic function f (x) = ax & # 178; + BX + C (a, B, C belong to R) satisfies the following conditions: ① When x belongs to R, the minimum value of F (x) is 0, and f (x-1) = f (- x-1) holds; ② When x belongs to (0,5), X ≤ f (x) ≤ 2 | X-1 | + 1 is constant (1) Find the value of F (1); (2) Find the analytic expression of F (x); (3) Find the largest real number m (M > 1), so that there is a real number T, as long as X belongs to [1, M], then f (x + T) ≤ x holds

It is known that the quadratic function f (x) = ax & # 178; + BX + C (a, B, C belong to R) satisfies the following conditions: (1) when x belongs to R, the minimum value of F (x) is 0 It is known that the quadratic function f (x) = ax & # 178; + BX + C (a, B, C belong to R) satisfies the following conditions: ① When x belongs to R, the minimum value of F (x) is 0, and f (x-1) = f (- x-1) holds; ② When x belongs to (0,5), X ≤ f (x) ≤ 2 | X-1 | + 1 is constant (1) Find the value of F (1); (2) Find the analytic expression of F (x); (3) Find the largest real number m (M > 1), so that there is a real number T, as long as X belongs to [1, M], then f (x + T) ≤ x holds

From condition 1, we can get a > 0, substitute f (x-1) = f (- x-1) into b = 2A, and then substitute the minimum value of 0 into f (x) = ax ^ 2 + 2aX + C, we can get C = a, that is, f (x) = ax ^ 2 + 2aX + a = a (x + 1) ^ 2
From condition 2, when x = 1, 1