已知函数f(x)=2sin(x-兀/6)cosx+2cos2x.求f(x)的单调递区间?急用

已知函数f(x)=2sin(x-兀/6)cosx+2cos2x.求f(x)的单调递区间?急用
已知函数f(x)=2sin(x-兀/6)cosx+2cos2x.求f(x)的单调递区间?急用

已知函数f(x)=2sin(x-兀/6)cosx+2cos2x.求f(x)的单调递区间?急用
f(x)=2sin(x-兀/6)cosx+2cos2x
=2(sinxcosπ/6-cosxsinπ/6)cosx+2cos2x
=√3sinxcosx-cos²x+2cos2x
=√3/2*sin2x-1/2(1+cos2x)+2cos2x
=√3/2*sin2x+3/2cos2x-1/2
=√3sin(2x+π/3)-1/2
由2kπ-π/2≤2x+π/3≤2kπ+π/2
得kπ-5π/12≤x≤kπ+π/12
增区间 [kπ-5π/12,kπ+π/12],k∈Z