已知函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240114289811247.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011428997567.png)
.
(1)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429012383.png)
為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429028463.png)
的極值點,求實數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429044283.png)
的值;
(2)當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429059471.png)
時,方程
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429090971.png)
有實根,求實數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429106299.png)
的最大值。
試題分析:(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240114292151066.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240114292311348.png)
. 1分
因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429246383.png)
為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429262495.png)
的極值點,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429293607.png)
. 2分
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429309689.png)
,解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429122370.png)
. 3分
又當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429340370.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429371680.png)
,從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429387627.png)
的極值點成立. 4分
(2)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429059471.png)
時,方程
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429418911.png)
可化為,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429449894.png)
.
問題轉(zhuǎn)化為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240114294651031.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429480563.png)
上有解,
即求函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429496767.png)
的值域. 7分
以下給出兩種求函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429527484.png)
值域的方法:
方法1:因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429543875.png)
,令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429558911.png)
,
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240114295901138.png)
, 9分
所以當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429605819.png)
,從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429621701.png)
上為增函數(shù),
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429636748.png)
,從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429652766.png)
上為減函數(shù), 10分
因此
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429668702.png)
.
而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429683393.png)
,故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429699687.png)
,
因此當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429168323.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429106299.png)
取得最大值0. 12分
方法2:因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429543875.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429777877.png)
.
設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429792867.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240114298701098.png)
.
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429902732.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429933637.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429948503.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429964688.png)
上單調(diào)遞增;
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429980640.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011430011628.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429948503.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011430073758.png)
上單調(diào)遞減;
因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011430089555.png)
,故必有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240114301041008.png)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240114301361339.png)
,
因此必存在實數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011430151899.png)
使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011430182596.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011430198861.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011430198765.png)
上單調(diào)遞減;
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011430214825.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011430229730.png)
上單調(diào)遞增;
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240114303231244.png)
上單調(diào)遞減;
又因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240114303381562.png)
,
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011430354803.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011430370529.png)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011430385499.png)
.
因此當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429168323.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824011429106299.png)
取得最大值0. 12分
點評:主要是考查了運用導(dǎo)數(shù)來判定函數(shù)單調(diào)性以及函數(shù)的 極值問題,通過利用函數(shù)的單調(diào)性放縮法來證明不等式,進而得到最值,屬于中檔題。
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