f(x)= 2 *(cos2x+ 1)/2+√3 * sin(2x)
=cos2x+√3*sin2x+a+ 1
= 2cos(2x-π/3)+a+ 1;
So T=2π/ω=π.
-π/6≤x≤π/3; -π/3≤2x≤2π/3; -2π/3≤2x-π/3≤π/3;
-1≤ 2cos (2x-π/3) ≤1,so a≤f(x)≤a+2.
So 2a+2 = 3 and a = 1/2.