(滿分12分)
已知函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002642991773.png)
.
(1)判斷并證明函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643007447.png)
的單調(diào)性;
(2)若函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643007447.png)
為奇函數(shù),求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643038265.png)
的值;
(3)在(2)的條件下,若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643053898.png)
對
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643069433.png)
恒成立,求實(shí)數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643085337.png)
的取值范圍.
(1)函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643007447.png)
在R上是增函數(shù)(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643116454.png)
(3)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643131532.png)
試題分析:(1) 任取
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643147573.png)
且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643163429.png)
∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643163429.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643225481.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643241638.png)
∴函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643007447.png)
在R上是增函數(shù) …………5分
(2)法1:∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643007447.png)
是奇函數(shù)∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643397487.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643116454.png)
…………8分
法2:∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643007447.png)
是奇函數(shù) ∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643428654.png)
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643459504.png)
得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643116454.png)
(3)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643053898.png)
即為
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643521657.png)
對
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643069433.png)
恒成立 …………10分
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643553325.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643568502.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240026435841350.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643599615.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824002643131532.png)
即為所求范圍 …………12分
點(diǎn)評:判定單調(diào)性可用定義可用導(dǎo)數(shù),不等式恒成立問題轉(zhuǎn)化為求函數(shù)最值問題
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