( 1).f( 1+x)=f( 1-x)

The symmetry axis is x= 1.

So -b/(2a)= 1.

b=-2a

Axe? +bx=x

Axe? +(b- 1)x=0

x[ax+(b- 1)]=0

x=0，x=-(b- 1)/a

Equal heel rule -(b- 1)/a=0

b= 1

a=-b/2=- 1/2

f(x)=-x？ /2+x

(2)

f(x)=-( 1/2)x^2+x=-( 1/2)(x- 1)^2+( 1/2)

When x= 1, the maximum value of f(x) is 1/2.

If x= 1, in the interval [m, n], then 3n= 1/2, and n= 1/6, then x < = 1/6, which is contradictory.

So: x= 1 is not in the interval [m, n]

When m<n< 1

F(x) monotonically increases in the interval [m, n]

So: 3m = f (m) =-(1/2) m2+m.

m^2+4m=0

And: 3n = f (n) =-(1/2) N2+n.

n^2+4n=0

So m and n are the two roots of the equation x 2+4x = 0.

x(x+4)=0

So: m=-4, n=0.

When 1

F(x) monotonically decreases in the interval [m, n]

So: 3m = f (n) =-(1/2) n 2+n.

n^2-2n+6m=0 - ( 1)

And: 3n = f (m) =-(1/2) m 2+m.

m^2-2m+6n=0 - (2)

(2)-( 1):

(m-n)(m+n-8)=0

Because m-n

n=8-m - (3)

(3) Replace (2) to obtain:

m^2-2m+48-6m=0

M 2-8m+48 = 0 has no solution.

To sum up:

m=-4，n=0

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