Question

已知f(x)=1-2x/x+1求单调区间并对其单调性证明

Answer 1

f(x)=(1-2x)/(x+1)
=(-2x-2+3)/(x+1)
=-2+3/(x+1)
f(x)是减函数
减区间是(-∞,-1)和(-1,+∞)

证明
设x1<x2，且x1,x2≠-1
f(x1)-f(x2)
=(1-2x1)/(x1+1)-(1-2x2)/(x2+1)
=[(1-2x1)(x2+1)-(1-2x2)(x1+1)]/[(x1+1)(x2+1)]
=3(x2-x1)/[(x1+1)(x2+1)]
∵x1<x2,
∴x2-x1>0
∴f(x1)-f(x2)>0
∴f(x)在(-∞,-1)和(-1,+∞)是减函数
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