設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848771283.png)
為實(shí)數(shù),函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848802936.png)
(Ⅰ)求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848818447.png)
的單調(diào)區(qū)間與極值;
(Ⅱ)求證:當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848834536.png)
且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848849393.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848865691.png)
(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848818447.png)
的單調(diào)遞減區(qū)間是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848912618.png)
,單調(diào)遞增區(qū)間是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848927621.png)
,極小值為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848958773.png)
;(Ⅱ) 見解析.
試題分析:(Ⅰ)直接根據(jù)導(dǎo)數(shù)和零的大小關(guān)系求得單調(diào)區(qū)間,并由單調(diào)性求得極值;(Ⅱ)先由導(dǎo)數(shù)判斷出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848974442.png)
在R內(nèi)單調(diào)遞增,說明對(duì)任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848990641.png)
,都有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849005615.png)
,而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849021491.png)
,從而得證.
試題解析:(1)解:由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849036925.png)
知,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849052819.png)
.
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849068535.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849083456.png)
.于是,當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849099266.png)
變化時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849114524.png)
和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849130491.png)
的變化情況如下表:
故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848818447.png)
的單調(diào)遞減區(qū)間是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848912618.png)
,單調(diào)遞增區(qū)間是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848927621.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848818447.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849083456.png)
處取得極小值,極小值為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848958773.png)
.
(2)證明:設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240218494111017.png)
,于是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849426940.png)
.
由(1)知,對(duì)任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849442424.png)
,都有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849458557.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848974442.png)
在R內(nèi)單調(diào)遞增.
于是,當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848834536.png)
時(shí),對(duì)任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848990641.png)
,都有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849005615.png)
,而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849021491.png)
,
從而對(duì)任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848990641.png)
,都有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849598539.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021849614785.png)
故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021848865691.png)
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