试题分析:(Ⅰ)证明对每一个
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848483523.png)
,存在唯一的
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848499679.png)
,满足
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848514593.png)
,只需证明两点,第一证
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848701500.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848717486.png)
上为单调函数,第二证,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848701500.png)
在区间
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848717486.png)
的端点的函数值异号,本题是高次函数,可用导数法判断单调性,而判断
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240228488111350.png)
的符号是,可用放缩法;(Ⅱ)由(Ⅰ)中的
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848514344.png)
构成数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848530475.png)
,判断数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848530475.png)
的单调性,由(Ⅰ)知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848701500.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848873535.png)
上递增,只需比较
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848889715.png)
的大小,由(Ⅰ)知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848514593.png)
,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848920650.png)
,而
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240228489352650.png)
,从而得到
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240228489511110.png)
,而
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848514593.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848982727.png)
,这样就可判断数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848530475.png)
的单调性;(Ⅲ)对任意
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848561538.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848577504.png)
满足(Ⅰ),试比较
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848592534.png)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848623348.png)
的大小,由(Ⅱ)知数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848639457.png)
单调递减,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022849060615.png)
,即比较
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022849076498.png)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848623348.png)
的大小,由(Ⅰ)知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848514593.png)
,写出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022849123521.png)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022849123616.png)
的式子,两式作差即可.本题函数与数列结合出题,体现学科知识交汇点的灵活运用,的确是一个好题,起到把关题的作用.
试题解析:(Ⅰ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240228491381272.png)
,显然,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022849154393.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022849169614.png)
,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848701500.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848873535.png)
上递增,又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240228492161054.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240228492323950.png)
,故存在唯一的
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022849247610.png)
,满足
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848514593.png)
;
(Ⅱ)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240228492791245.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240228494972586.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240228495131252.png)
,由(Ⅰ)知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848701500.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848873535.png)
上递增,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022849559472.png)
,即数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848639457.png)
单调递减;
(Ⅲ) 由(Ⅱ)数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848639457.png)
单调递减,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022849060615.png)
,而
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240228496221219.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240228496372562.png)
,两式相减:并结合
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022849653605.png)
,以及
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022849247610.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240228496841438.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240228497151426.png)
,所以有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824022848655705.png)
.