Question

已知函数f(x)=2x2-1.用定义证明f(x)在(-无穷,0]上是减函数

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Answer 1

任取0<x1<x2<1
f(x1)-f(x2)
=(x1 1/x1)-(x2 1/x2)
=x1-x2 1/x1-1/x2
=x1-x2 (x2-x1)/(x1x2)
=(x1-x2)[1-1/(x1x2)]
因0<x1<x2<1
x1-x2<0
0<x1x2<1
1/(x1x2)>1
1-1/(x1x2)<0
所以(x1-x2)[1-1/(x1x2)>0
所以f(x1)>f(x2)
所以f(x)在(0,1)上是减函数