高一数学题

来源:百度知道 编辑:互助问答吧 时间:2020/08/07 08:17:57
已知函数f(x)=(cosx)^4-2sinxcosx-(sinx)^4
(1)求f(x)的最小正周期
(2)当x属于[0,派/2]时,求f(x)的最小值以及取得最小值时x的集合

f(x)=(cosx)^4-2sinxcosx-(sinx)^4 =(cosx)^4-(sinx)^4 -2sinxcosx
=((cosx)^2+(sinx)^2)((cosx)^2-(sinx)^2)-2sinxcosx
=((cosx)^2-(sinx)^2)-2sinxcosx=cos2x-sin2x=cos2x-cos(90°-2x)
=-2sin45°sin(45°-2x)
1)T=360°/2=180°=π
2)sin(45°-2x)=1时f(X)有最小值
45-2x=90,x=22.5°=π/8
x∈(π/8+nπ),n整数

(1)f(x)=[(cosx)^2+(sinx)^2][(cosx)^2-(sinx)^2]-sin2x
=-sin2x+cos2x=√2sin(2x-π/4)
T=2π/2=π
(2)f(x)=√2sin(2x-π/4)
f(x)min=-√2
sin(2x-π/4)=-1
2x-π/4=-π/2+kπ
x=-π/8+kπ/2
x=3π/8
x∈{3π/8}