已知函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645046819.png)
(1)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645062375.png)
,試確定函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645093447.png)
的單調(diào)區(qū)間;
(2)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645109424.png)
,且對(duì)于任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645124424.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645140586.png)
恒成立,試確定實(shí)數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645171312.png)
的取值范圍;
(1)詳見解析(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645187481.png)
.
試題分析:(1)求出函數(shù)的導(dǎo)數(shù),只要解導(dǎo)數(shù)的不等式即可,根據(jù)導(dǎo)數(shù)與0的關(guān)系判斷函數(shù)的單調(diào)性;
(2)函數(shù)f(|x|)是偶函數(shù),只要f(x)>0對(duì)任意x≥0恒成立即可,等價(jià)于f(x)在[0,+∞)的最小值大于零.
試題解析:解:(1)由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645062375.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645234611.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645249627.png)
.
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645265570.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645280360.png)
,故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645093447.png)
的單調(diào)遞增區(qū)間是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645327507.png)
,
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645343560.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645358340.png)
,故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645093447.png)
的單調(diào)遞減區(qū)間是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645405492.png)
. 4
(2)由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645436669.png)
可知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645452487.png)
是偶函數(shù).
于是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645140586.png)
對(duì)任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645124424.png)
成立等價(jià)于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645514538.png)
對(duì)任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645530394.png)
成立.
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645561724.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645577461.png)
.
①當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645592531.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240446456241019.png)
.
此時(shí)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645093447.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645655505.png)
上單調(diào)遞增.
故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645670746.png)
,符合題意.
②當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645686618.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645702485.png)
.
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645717266.png)
變化時(shí)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645748590.png)
的變化情況如下表:
由此可得,在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645655505.png)
上,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645936864.png)
.
依題意,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645951568.png)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645967582.png)
.
綜合①,②得,實(shí)數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645171312.png)
的取值范圍是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824044645187481.png)
.
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