f(x)=sinx^2+2倍根号3sinxcosx+3cosx^2的单调增区间

题目

f(x)=sinx^2+2倍根号3sinxcosx+3cosx^2的单调增区间

答案

f(x)=sin²x+2√3sinxcosx+3cos²x
=1+2cos²x+√3sin2x
=2+(2cos²x-1)+√3sin2x
=cos2x+√3sin2x+2
=2(sinπ/6cos2x+cosπ/6sin2x)+2
=2sin(2x+π/6)+2
当2kπ-π/2≤2x+π/6≤2kπ+π/2
f(x)是增函数
即f(x)的单调增区间为[kπ-π/3,kπ+π/6]