已知函数f(x)=2sin²x/4cosπ/4-2√3sin²x/4+√3,求函数的最小正周期及最值.

f(x)=2sinx/4cosx/4-2√3sin²x/4+√3
=sin(x/2)+√3[1-2sin²(x/4)]
=sin(x/2)+√3cos(x/2)
=2[1/2*sin(x/2)+√3/2*cos(x/2)]
=2sin(x/2+π/3)
最小正周期T=2π/(1/2)=4π
当x/2+π/3=2kπ+π/2,k∈Z
即x=4kπ+π/3,k∈Z时,f(x)取得最大值2
当x/2+π/3=2kπ-π/2,k∈Z
即x=4kπ-5π/3,k∈Z时,f(x)取得最大值-2