函数f(x)=sin^4x+2根号3sinxcosx-cos^4x的值域为
问题描述：
函数f(x)=sin^4x+2根号3sinxcosx-cos^4x的值域为
答
f(x)=sin^4x+2根号3sinxcosx-cos^4x
=(sin^2x+cos^2x)(sin^2x-cos^2x)+√3*sin2x
=-cos2x+√3sin2x
=2*(√3/2*sin2x-1/2*cos2x)
=2sin(2x-∏/6).
当sin(2x-∏/6)=1时,f(x)取最大值,f(x)=2,
当sin(2x-∏/6)=-1时,f(x)取最小值,f(x)=-2.
函数f(x)=sin^4x+2根号3sinxcosx-cos^4x的值域为:[-2,2]
答
y=sin^4(x)+2√3(sinxcosx)-cos^4(x) =sin^4(x)-cos^4(x)+2√3sinxcosx =(sin^2x+cos^2x)(sin^2x-cos^2x)+2√3(sinxcosx) =2√3(sinxcosx)-cos2x =√3sin2x-cos2x =2sin(2x-π/6)∴值域是[-2,2]