f(x)=2cos²x+2√3sinxcosx
=1+cos(2x)+√3sin(2x)
=2[(√3/2)sin(2x)+(1/2)cos(2x)]+1
=2sin(2x+π/6)+1
当sin(2x+π/6)=1时,f(x)有最大值f(x)max=2×1+1=3
此时,2x+π/6=2kπ+π/2 (k∈Z)
x=kπ+π/6 (k∈Z)
所求自变量x的集合为{x|x=kπ+π/6,k∈Z}