试题分析:(Ⅰ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616598790.png)
. 由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616598517.png)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616613395.png)
,此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616629672.png)
.
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616644568.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616660570.png)
,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616317447.png)
在区间
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616691452.png)
上单调递增;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616707641.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616738560.png)
,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616317447.png)
在区间
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616754535.png)
上单调递减.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616769195.png)
函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616317447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616301367.png)
处取得极大值,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616613395.png)
. 3分
(Ⅱ)令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240106168321892.png)
, 4分
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240106168781136.png)
.函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616317447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616457598.png)
上可导,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616769195.png)
存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616941637.png)
,使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240106169721056.png)
.又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616988679.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240106170031838.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617034598.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617050593.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617066484.png)
单调递增,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617081746.png)
;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617097613.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617112590.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617066484.png)
单调递减,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617159775.png)
;
故对任意
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616457598.png)
,都有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616473594.png)
. 8分
(Ⅲ)用数学归纳法证明.
①当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617222414.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617237621.png)
,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617253476.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617268495.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617300831.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616769195.png)
由(Ⅱ)得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616473594.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240106173622516.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616769195.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617222414.png)
时,结论成立. 9分
②假设当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617487648.png)
时结论成立,即当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617502739.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240106175021962.png)
. 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617518468.png)
时,设正数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617534666.png)
满足
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617549777.png)
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617565820.png)
,
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617596630.png)
,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617612657.png)
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240106176272509.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240106176434194.png)
13分
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616769195.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010617518468.png)
时,结论也成立.
综上由①②,对任意
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616520435.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824010616535490.png)
,结论恒成立. 14分
点评:近几年新课标高考对于函数与导数这一综合问题的命制,一般以有理函数与半超越(指数、对数)函数的组合复合且含有参量的函数为背景载体,解题时要注意对数式对函数定义域的隐蔽,这类问题重点考查函数单调性、导数运算、不等式方程的求解等基本知识,注重数学思想(分类与整合、数与形的结合)方法(分析法、综合法、数学归纳法)的运用.把数学运算的“力量”与数学思维的“技巧”完美结合.