Question

求函数y=1/2cos^2x+根号3/2sinxcosx+1的最大值.并写出函数取最大值时的自变量x值的集合

Answer 1

解

y=1/2cos²x+√3/2sinxcosx+1

=1/4(2cos²x-1)+√3/4(2sinxcosx)+5/4

=1/4cos2x+√3/4sin2x+5/4

=1/2[1/2cos2x+√3/2sin2x]+5/4

=1/2[sin2xcosπ/6+cos2xsinπ/6]+5/4

=1/2sin(2x+π/6)+5/4

当2x+π/6=π/2+2kπ

即x=π/6+kπ时

y取得最大值为：1/2+5/4=7/4

∴x的集合为：{x=π/6+kπ,k∈z}