试题与答案

已知定义在实数集R上的偶函数f(x)在区间[0,+∞)上是单调增函数. (1)求

题型:解答题

题目:

已知定义在实数集R上的偶函数f(x)在区间[0,+∞)上是单调增函数.

(1)求证:函数f(x)在区间(-∞,0]上是单调减函数

(2)若f(1)<f(lgx),求x的取值范围.

答案:

(1)证明:设x1<x2≤0,则-x1>-x2≥0

∵f(x)在区间[0,+∞)上是单调增函数.

∴f(-x1)>f(-x2

又定义在实数集R上的偶函数f(x)

∴f(-x1)=f(x1),f(-x2)=f(x2),f(x1)>f(x2

∴函数f(x)在区间(-∞,0]上是单调减函数

(2)当0<x≤1时,lgx<0

由f(1)<f(lgx)得f(-1)<f(lgx),函数f(x)在区间(-∞,0]上时单调减函数

-1>lgx,0<x<

1
10

当x≥1时,lgx>0

由f(1)<f(lgx),f(x)在区间[0,+∞)上是单调增函数

∴lgx>1,x>10

综上所述,x的取值范围是(0,

1
10
][10,+∞)

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