已知函数f(X)等于根号2*sin(2X+π/4),若任意X属于[0,π/2],f(X)小于m,求实数m的取值范围

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已知函数f(X)等于根号2*sin(2X+π/4),若任意X属于[0,π/2],f(X)小于m,求实数m的取值范围
已知函数f(X)等于根号2*sin(2X+π/4),若任意X属于[0,π/2],f(X)小于m,求实数m的取值范围

已知函数f(X)等于根号2*sin(2X+π/4),若任意X属于[0,π/2],f(X)小于m,求实数m的取值范围
∵x∈[0,π/2] ∴2x∈[0,π] ∴2X+π/4∈[π/4,5π/4] ∴sin(2X+π/4)∈[﹣√2/2,√2/2]
∴f(x)=√2sin(2X+π/4)∈[﹣1,1]
∵f(x)<m
∴m>1

f(X)=√2sin(2X+π/4),
∵x∈[0,π/2],
2x+π/4∈[π/4,5π/4]
∴sin(2X+π/4)∈[﹣√2/2,1]
2*sin(2X+π/4)∈[﹣√2,2]
∵f(X)小于m,
∴m>2

(√2,+∞)