已知x属于[45°,90°],且f(x)=2sin^2(x+45°)-根号3cos2x,求f(x)的最值

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已知x属于[45°,90°],且f(x)=2sin^2(x+45°)-根号3cos2x,求f(x)的最值已知x属于[45°,90°],且f(x)=2sin^2(x+45°)-根号3cos2x,求f(x

已知x属于[45°,90°],且f(x)=2sin^2(x+45°)-根号3cos2x,求f(x)的最值
已知x属于[45°,90°],且f(x)=2sin^2(x+45°)-根号3cos2x,求f(x)的最值

已知x属于[45°,90°],且f(x)=2sin^2(x+45°)-根号3cos2x,求f(x)的最值
f(x)
=2sin²(x+45)-√3cos2x
=[1-cos2(x+45)]-√3cos2x
=1-cos(2x+90)-√3cos2x
=1-sin(-2x)-√3cos2x
=1+sin2x-√3cos2x
=1+2[(1/2)sin2x-(√3/2)cos2x]
=1+2[cos60sin2x-sin60cos2x]
=1+2sin(2x-60)
因为45≤x≤90
所以30≤2x-60≤120
1/2≤sin(2x-60)≤1
2≤1+2sin(2x-60)≤3
即f(x)的值域是[2,3]
f(x)的最小值是2,最大值是3