题目内容
(本题13分)
已知函数
.
(1)当
时,求
的单调区间;
(2)若
在
单调增加,在
单调减少,证明:
<6.
已知函数
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181835879701.gif)
(1)当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181835911277.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181835926270.gif)
(2)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181835926270.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836051553.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836082552.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836098271.gif)
解:(1)当![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/2014082318183611365.gif)
时,
,故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836160968.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836176492.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836269573.gif)
当![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836269297.gif)
当![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836472648.gif)
从而
单调减少.----(6分)
(2)![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231818364881668.gif)
由条件得:![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231818365031003.gif)
从而![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836519791.gif)
因为![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836535595.gif)
所以![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836550934.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836581758.gif)
将右边展开,与左边比较系数得,![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836597576.gif)
故![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/2014082318183661372.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836628896.gif)
又![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836659986.gif)
由此可得
于是
--------------------(13分)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/2014082318183611365.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181835911277.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836145663.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836160968.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836176492.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836269573.gif)
当
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836269297.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836301656.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836472648.gif)
从而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231818364881136.gif)
(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231818364881668.gif)
由条件得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231818365031003.gif)
从而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836519791.gif)
因为
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836535595.gif)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836550934.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836581758.gif)
将右边展开,与左边比较系数得,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836597576.gif)
故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/2014082318183661372.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836628896.gif)
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836659986.gif)
由此可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836691258.gif)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823181836706334.gif)
略
![](http://thumb.zyjl.cn/images/loading.gif)
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