已知函数y=1/2cos平方x+[(根号3)/2]sinxcosx+1,x属于R

1/2=sin(π/6),√3/2=cos(π/6),因此可对表达式化简：y=(1/2)(cosx)^2+(√3/2)sinxcosx+1=cosx[sin(π/6)cosx+cos(π/6)sinx]+1=sin(x+π/6)cosx+1 ………………………………………………………（1）sin(2x+π/6)=sin(x+π/6+x)=sin(x+π/6)cosx+cos(x+π/6)sinx ………（2）1/2=sin(π/6)=sin(x+π/6-x)=sin(x+π/6)cosx-cos(x+π/6)sinx ………（3）（2）+（3）可得：sin(x+π/6)cosx=[sin(2x+π/6)]/2+1/4 ……………（4）把（4）代入（1）继续化简：sin(x+π/6)cosx+1=[sin(2x+π/6)]/2+1/4+1=[sin(2x+π/6)]/2+5/4因此：y=[sin(2x+π/6)]/2+5/4（1）y取最大值时,sin(2x+π/6)=1,即2x+π/6=2kπ+π/2,求得x=kπ+π/6（k∈Z）,因此所求x的集合为：{x|x=kπ+π/6（k∈Z）}（2）由函数表达式y=[sin(2x+π/6)]/2+5/4可知变换顺序：sinx → sin(x+π/12) → sin[2(x+π/12)]=sin(2x+π/6) → [sin(2x+π/6)]/2 → [sin(2x+π/6)]/2+5/4即将函数y=sinx的图像先整体左移π/12个单位,然后横向压缩一倍（即左右压缩）,之后纵向压缩一倍（即上下压缩）,最后整体上移5/4个单位,就可得到题设函数的图像。或者：sinx → sin2x → sin(2x+π/6) → [sin(2x+π/6)]/2 → [sin(2x+π/6)]/2+5/4即将函数y=sinx的图像先横向压缩一倍,然后整体左移π/12个单位,之后纵向压缩一倍,最后整体上移5/4个单位,也可得到题设函数的图像。