f(3-x) =f(x)
a(3-x)^2+b(3-x) + c = ax^2 +bx +c
compare coef. of x
=> -6a-b =b
a = -b/3 (1)
f(1) =0
=> a+b+c =0
-b/3+b+c =0
c = -2b/3 (2)
f(x)≥ a/4-1/2 ( for all x )
ax^2+bx+c ≥ a/4-1/2
a(x+ b/(2a))^2 + (c-b^2/(4a)) ≥ a/4-1/2 ( for all x )
=> c-b^2/(4a) = a/4-1/2 (3)
Sub (1),(2) into (3)
-2b/3 - b^2/(-4b/3) = (-b/3)/4 - 1/2
-2b/3+3b/4= -4b/3 -1/2
17b/12 = -1/2
b = -6/17
=> a =2/17, c= 4/17
f(x) = (2/17)x^2 - (6/17)x + 4/17