试题分析:(1)判定函数的奇偶性,首先判定定义域是否关于原点对称,定义域为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237064588.png)
关于原点对称,其次研究
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237079495.png)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237095502.png)
的相等或相反的关系:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240552371111892.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240552371421117.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055236970907.png)
为偶函数,(2)由于函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237079495.png)
为偶函数,所以只需证明
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237189393.png)
时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055236986591.png)
,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237189393.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237251724.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237267687.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237189393.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240552373131023.png)
恒成立,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237329386.png)
时,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237345409.png)
,由(1)可知:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237360683.png)
,综上所述,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055236986591.png)
在定义域内恒成立(3)恒成立问题一般利用变量分离法转化为最值问题.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237017922.png)
恒成立对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237001517.png)
恒成立,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240552374381089.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237469936.png)
,令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237485944.png)
可证
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237485944.png)
在[1,3]上为减函数 ∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240552375321203.png)
对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237001517.png)
恒成立 ∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237563646.png)
,所以m的取值范围是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237048693.png)
.
试题解析:解:(1)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055236970907.png)
为偶函数,证明如下:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055236970907.png)
定义域为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237064588.png)
关于原点对称,
对于任意
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237641653.png)
有: 2分
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240552376721928.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240552377351171.png)
成立
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055236970907.png)
为偶函数 5分
(2)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055236970907.png)
定义域为:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237064588.png)
,
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237189393.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237251724.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237267687.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237189393.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240552373131023.png)
恒成立, 7分
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237329386.png)
时,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237345409.png)
,由(1)可知:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237360683.png)
9分
综上所述,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055236986591.png)
在定义域内恒成立 10分
(3)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237017922.png)
恒成立对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237001517.png)
恒成立,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240552374381089.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237469936.png)
,令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237485944.png)
证明
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237485944.png)
在[1,3]上为减函数(略)(不证明单调性扣2分)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240552381091222.png)
对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237001517.png)
恒成立 12分
∴
所以m的取值范围是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824055237048693.png)
14分