(1)f（x）=2cosx*cosx+2sinx*√3cosx
=2cosx^2+√3sin2x
=1+cos2x+√3sin2x
=√3sin2x+cos2x+1
=2sin(2x+30°)+1
最小正周期T=2π/2=π
f（x）最大值=2+1=3
f（x）最小值=-2+1=-1
所以值域为-1≤f（x）≤3
(2)f（C）=2sin(2C+30°)+1=3
2sin(2C+30°)=2
sin(2C+30°)=1
2C+30°=90°
所以C=30°
三角形ABC的面积=1/2*ab*sinC
=1/2*2√3*1/2
=√3/2