It is known that the quadratic function f (x) = ax ^ 2 + BX satisfies f (2) = 0 and the equation f (x) = x has equal roots

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a=-0.5,b=1

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1. It is known that the quadratic function f (x) = ax ^ 2 + BX (AB ∈ R, a ≠ 0) satisfies f (- x + 5) = f (x-3) and the equation f (x) = x has equal roots; It is known that the quadratic function f (x) = ax ^ 2 + BX (AB ∈ R, a ≠ 0) satisfies f (- x + 5) = f (x-3) and the equation f (x) = x has equal roots. (1) find the analytic expression of F (x); (2) whether there are real numbers m, n (m < n), so that the domain of definition and value of F (x) are [M, n] and [3M, 3N]? If there are, find the value of M, N; if not, explain the reason
2. It is known that the quadratic function f (x) = ax + BX (a, B are familiar, and a ≠ 0) satisfies the following conditions: F (- x + 5) = f (x-3), and the equation f (x) = x has equal roots (1) The expression of finding f (x) (2) whether there is a teacher's uncle m, n (m) or not
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10. X divided by (1-radical x + 1) (x is greater than or equal to negative 1. X is not equal to 0)
11. If the quadratic function f (x) = AX2 + BX + C, if f (0) = 0 and f (x + 1) = f (x) + X + 1, then f (x)=
12. Given the quadratic function f (x) = AX2 + BX + C, f (2) = 0, f (- 5) = 0, f (0) = 1, find the quadratic function
13. Quadratic function, f (x) = AX2 + BX + C, f F (x) = AX2 + BX + C, f (x) = f (2-x), f (0) = 3, f (1) = 2.1. Find the analytic formula of function. 2. F (x) belongs to the maximum value of [- 1,2] in X
14. It is known that the quadratic function f (x) = AX2 + BX + C (a > 0) has two different common points with the X axis If f (c) = 0 and 0 < x < C, f (x) > 0 1) Compare 1 / a with C 2) Proof - 2 < B < 1
15. Given the quadratic function f (x) = AX2 + BX + C (C ≠ 0) (1) if A.B.C and f (1) = 0, it is proved that the image of F (x) has two intersections with the X axis; (2) if Changshu x1 If x 2 ∈ R, and x 1, x 2, f (x 1) ≠ f (x 2), we prove that the equation f (x) = 1 / 2 [f (x 1) + F (x 2)] must have a root belonging to (x 1, x 2)
16. 7. Given the quadratic function f (x) = AX2 + BX + C, G (x) = λ ax + B (λ≥ 1), when | x | ≤ 1, | f (x) | ≤ 1. (1) it is proved that | a | ≤ 2 (2) Let f (0), f (1), f (- 1) denote g (1), G (- 1) (3) When | x | ≤ 1, it is proved that | g (x) | ≤ 2 λ
17. It is known that the derivative f '(x) ≤ f (x) B of quadratic function f (x) = AX2 + BX + C belongs to R. it is proved that f (x) is less than or equal to (x + C) ^ 2 when x ≥ 0
18. If f (x) = AX2 + BX (a ≠ 0) satisfies 1 ≤ f (- 1) ≤ 2, 2 ≤ f (1) ≤ 5, then the value range of F (- 3) is______ ．
19. If f (x) = AX2 + BX, and 1 ≤ f (1) ≤ 2,2 ≤ f (1) ≤ 4, find the range of F (2)
20. Let f (x) = alnx-1 / 2x ^ 2 + BX find the solution set of the inequality f (x) > F (1)