函数f(x)=4sinxsin^2(45°+2分之x)+cos2x,x∈[π/6,2π/3],都有|f(x)-m|

函数f(x)=4sinxsin^2(45°+2分之x)+cos2x,x∈[π/6,2π/3],都有|f(x)-m|
函数f(x)=4sinxsin^2(45°+2分之x)+cos2x,x∈[π/6,2π/3],都有|f(x)-m|

函数f(x)=4sinxsin^2(45°+2分之x)+cos2x,x∈[π/6,2π/3],都有|f(x)-m|
∵f(x)=4sinxsin²(π/4+x/2)+cos(2x)
=2sinx[2sin²(π/4+x/2)]+cos(2x)
=2sinx[1-cos(2(π/4+x/2))]+cos(2x)
=2sinx[1-cos(π/2+x)]+cos(2x)
=2sinx(1+sinx)+1-2sin²x
=2sinx+1
∵x∈[π/6,2π/3],
f(π/6)=2sin(π/6)+1=2
f(π/2)=2sin(π/2)+1=3
f(2π/3)=2sin(2π/3)+1=1+√3
∴f(x)最大值是f(π/2)=3,最小值是f(π/6)=2.
|f(x)-m|