y = cos²x+√3sinxcosx+1/2
= 1/2 (cos2x+1) + √3/2 sin2x + 1/2
= √3/2 sin2x + 1/2 cos2x + 1
= sin2xcosπ/6 + cos2xsinπ/6 + 1
= sin(2x+π/6) + 1
最小正周期=2π/2=π
2x+π/6∈（2kπ-π/2,2kπ+π/2）时单调增；2x+π/6∈（2kπ+π/2,2kπ+3π/2）时单调减
单调增区间：（kπ-π/3,kπ+π/6）,其中k∈Z
单调减区间：（kπ+π/6,kπ+2π/3）,其中k∈Z
2x+π/6=2kπ+π/2时,函数最大值=1+1=2
此时x = kπ+π/6,其中k∈Z