Question

集合M由滿足以下條件的函數(shù)f(x)組成:對任意的x₁、x₂∈［-1,1］時,都有|f(x₁)-f(x₂)|≤4|x₁-x₂|.對于兩個函數(shù)f₁(x)=x²-2x+5,f₂(x)=,以下關系成立的是

A.f₁(x)∈M,f₂(x)∈M                           B.f₁(x)M,f₂(x)M

C.f₁(x)M,f₂(x)∈M                            D.f₁(x)∈M,f₂(x)M

Answer 1

答案:D

解析：|f₁(x₁)-f₁(x₂)|=|x₁²-2x₁-x₂²+2x₂|=|(x₁-x₂)(x₁+x₂-2)|=|x₁-x₂||x₁+x₂-2|.∵x₁,x₂∈［-1,1］,

∴0≤|x₁+x₂-2|≤4.∴|f₁(x₁)-f₁(x₂)|≤4|x₁-x₂|,即f₁(x)∈M;而當x₁=,x₂=時,

|f₂(x₁)-f₂(x₂)|=,4|x₁-x₂|=4()=＜,

故f₂(x)M.∴選D.