已知函数f(x)=sin^2x+√3xcosx+2cos^2x,x属于R求f(x)的最小正周期和单调增区间.(主要写出化简的过程,

Question

已知函数f(x)=sin^2x+√3xcosx+2cos^2x,x属于R
求f(x)的最小正周期和单调增区间.
(主要写出化简的过程,

Answer 1

f(x)=√3sinxcosx+cos²x+(sin²x+cos²x)
=√3/2*(2sinxcosx)+(1+cos2x)/2+1
=√3/2*sin2x+1/2*cos2x+3/2
=sin2xcosπ/6+cos2xsinπ/6+3/2
=sin(2x+π/6)+3/2
所以T=2π/2=π
sinx增区间是(2kπ-π/2,2kπ+π/2)
所以这里单调增则2kπ-π/2