Question

判断函数f(x)=x+1/x在区间（-1，0）上的单调性并给出证明

Answer 1

函数在区间(-1，0)上是减函数，证明如下：
证：
设定义域上-1<x1<x2<0
f(x2)-f(x1)
=x2+1/x2-x1-1/x1
=(x2-x1)-(x2-x1)/(x1x2)
=(x2-x1)[1-1/(x1x2)]
=(x2-x1)(x1x2-1)/(x1x2)
x2>x1 x2-x1>0
-1<x1<0 -1<x1x2<0
0<x1x2<1
x1x2-1<0
(x2-x1)(x1x2-1)/(x1x2)<0
f(x2)<f(x1)
函数是减函数。