题目内容
设定义在
上的函数
,满足当
时,
,且对任意
,有
,![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156220562.png)
(1)解不等式![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156236768.png)
(2)解方程![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240201562511206.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156127299.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156127495.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156142393.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156158568.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156173532.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156205866.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156220562.png)
(1)解不等式
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156236768.png)
(2)解方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240201562511206.png)
(1)先证
,且单调递增,
;(2)
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156267591.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156283644.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156298367.png)
试题分析:(1)先证
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156267591.png)
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156329890.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156142393.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156158568.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156376533.png)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240201563921529.png)
假设存在某个
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156407496.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156423623.png)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240201564391361.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156267591.png)
任取
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156548563.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156563429.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156579528.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156579686.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156595686.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156610921.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156626833.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156641944.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156657433.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156127495.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156283644.png)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156220562.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156719574.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156735596.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156735955.png)
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156751548.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020156766587.png)
点评:难题,涉及抽象不等式解法问题,往往利用函数的奇偶性、单调性,将抽象问题转化成具体不等式组求解,要注意函数的定义域。抽象函数问题,往往利用“赋值法”,通过给自变量“赋值”,发现结论,应用于解题。本题较难,构造结构形式,应用已知条件,是解答本题的一大难点。
![](http://thumb2018.1010pic.com/images/loading.gif)
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