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

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

(1) From F (- x + 5) = f (x-3), the,
The image of F (x) is symmetric with respect to x = (5-3) / 2,
-b/2a=1.①
The equation f (x) = x has equal roots,
That is, ax ^ 2 + BX = x,
Ax ^ 2 + (B-1) x = 0 has equal root 0,
x=(1-b)/a=0.②
From (1) to (2)
a=-1/2,b=1.
So f (x) = - 1 / 2 · x ^ 2 + X
(2)
There are three types of cases
1. In the monotone increasing interval x = 1 / 2
Solving equation rent
3y=-(1/2)x^2+x
3x=-(1/2)y^2+y
unsolvable
3. Including the maximum point (1 / 2,3 / 8)
Then 3N = 3 / 8, n = 1 / 8
m