Then f(x)=f(-x)

that is

(a-2) x2+(a-1) x+3 = (a-2) x2-(a-1) x+3 holds.

Therefore, the coefficient of the first term is 0.

a- 1=0

a= 1

Then f(x)=-x2+3.

F(x) is a parabola with a downward opening, and the symmetry axis is x=0.

So when x=0,

The maximum value of F(x) is 3.