Question

已知函数f(x)=sinx/2cosx/2+根号3sin2x/2+根号3/2,求函数值域和单调增区间

Answer 1

原题是:已知函数f(x)=sin(x/2)cos(x/2)+(√3)(sin(x/2))^2+(√3)/2,求函数值域和单调增区间.
f(x)=sin(x/2(cos(x/2)+(√3)(sin(x/2))^2+(√3)/2
=(1/2)sinx+(√3).(1/2)(1-cosx)+(√3)/2
=(1/2)sinx-(√3)/2cosx+(√3)
=sin(x-π/3)+(√3)
值域是[-1+√3,1+√3].
x在单增区间中:2kπ-π/2≤x-π/3≤2kπ+π/2,k∈Z
2kπ-π/6≤x≤2kπ+5π/6,k∈Z
所以f(x)的单调增区间是[2kπ-π/6,2kπ+5π/6],k∈Z
希望对你有点帮助!