求函数的最值,周期（1）y=sinx/4cosx/4-1（2）y=sinx+2cosx（3）y=2cosxsin（x+π/3）-根号3sin²x+sinxcosx

Question

求函数的最值,周期（1）y=sin x/4cos x/4-1（2）y=sinx+2cosx
（3）y=2cosxsin（x+π/3）-根号3sin²x+sinxcosx

Answer 1

1.
y=sin(x/4)cos(x/4)-1=(1/2)sin(x/2) -1
最小正周期T=2π/(1/2)=4π
sin(x/2)=1时,y有最大值ymax=1/2 -1=-1/2；sin(x/2)=-1时,y有最小值ymin=-1/2-1=-3/2
2.
y=sinx+2cosx=√5sin(x+a),其中,tana=2
最小正周期T=2π/1=2π
sin(x+a)=1时,y有最大值ymax=√5；sin(x+a)=-1时,y有最小值ymin=-√5
3.
y=2cosxsin(x+π/3)-√3sin²x+sinxcosx
=2cosx[sinxcos(π/3)+cosxsin(π/3)]-√3sin²x+sinxcosx
=2cosx[(1/2)sinx+(√3/2)cosx]-√3sin²x+sinxcosx
=sinxcosx+√3cos²x-√3sin²x+sinxcosx
=√3(cos²x-sin²x)+2sinxcosx
=√3cos(2x)-sin(2x)
=2[(√3/2)cos2x-(1/2)sin(2x)]
=2cos(2x+π/6)
最小正周期T=2π/2=π
cos(2x+π/6)=1时,y有最大值ymax=2；cos(2x+π/6)=-1时,y有最小值ymin=-2