Question

急！！已知函数f(x)=(sinx-cosx)sinx,x∈R,则f(x)取最小值时,x的集合为

Answer 1

f(x)=(sinx)^2-cosxsinx=(1-cos2x)/2-(sin2x)/2
f(x)=[1-cos2x-sin2x]/2=1/2-(cos2x+sin2x)/2
f(x)最小 (cos2x+sin2x)/2 最大 即cos2x+sin2x最大
cos2x+sin2x=√2sin(2x+π/4) 2x+pi/4=pi/2+2kpi
2x=pi/4+2kpi x=pi/8+kpi

Answer 2

f(x)=sinx^2-cosxsinx=1/2(1-cos2x)-1/2sin2x=1/2-1/2(sin2x+cos2X)=1/2-√2/2sin(2x+π/4)
当sin(2x+π/4)=1时，即2x+π/4=2kπ+π/2,x=kπ+π/8时,最小值为1/2-√2/2