试题分析:解:(Ⅰ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958514609.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958530863.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958561548.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958577338.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958592718.png)
,又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958608487.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958624505.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958639352.png)
综上可知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958670582.png)
,定义域为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958686266.png)
>0,
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958702481.png)
<0 得 0<
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958686266.png)
<
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958733303.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958421447.png)
的单调减区间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958764533.png)
……………6分
(Ⅱ)先证
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240039587801209.png)
即证
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240039588111281.png)
即证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240039588261205.png)
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958826524.png)
,∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958436313.png)
>0,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958452310.png)
>0 ,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958873267.png)
>0,即证
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958889981.png)
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240039589041213.png)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240039589201420.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240039589362276.png)
① 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958951453.png)
>
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958967335.png)
,即0<
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958873267.png)
<1时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958998656.png)
>0,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003959014496.png)
>0
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003959154467.png)
在(0,1)上递增,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003959154467.png)
<
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003959185449.png)
=0,
② 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958951453.png)
<
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958967335.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958873267.png)
>1时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958998656.png)
<0,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003959014496.png)
<0
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003959154467.png)
在(1,+∞)上递减,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003959154467.png)
<
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003959185449.png)
=0,
③ 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958951453.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958967335.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958873267.png)
=1时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003959154467.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003959185449.png)
=0
综合①②③知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003959513580.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824003958889981.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240039587801209.png)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240039595441719.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240039595601104.png)
综上可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240039595751206.png)
……………14分
点评:对于导数在研究函数中的运用,关键是利用导数的符号判定单调性,进而得到极值,和最值, 证明不等式。属于中档题。