Question

已知函数f(x)=2sin^2(Π/4+x)-√3cos2x,x∈[Π/4, Π/2].(1)求f(x)的最大值和最小

已知函数f(x)=2sin^2(Π/4+x)-√3cos2x,x∈[Π/4, Π/2].(1)求f(x)的最大值和最小值；(2)若不等式|f(x)-m|<2在x∈[Π/4, Π/2]上恒成立，求实数m的取值范围

Answer 1

∵f(x)=2sin²(π/4+x)-√3cos2x＝1－[1－2sin²(π/4+x)]-√3cos2x＝1－cos(π/2+2x)－√3cos2x
＝1＋sin2x－√3cos2x＝1＋2sin(2x－π/3)
(1)∵x∈[π/4, π/2] ∴2x∈[π/2,π] 2x－π/3∈[π/6,2π/3] ∴sin(2x－π/3)∈[1/2,1]
∴f(x)的最大值＝1＋2×1＝3 最小值＝1＋2×1/2＝2
(2)∵|f(x)－m|＜2 ∴﹣2＜f(x)－m＜2 ∴﹣2＋f(x)＜m＜2＋f(x)
∵2＜f(x)＜3 ∴1＜m＜4