试题分析:(Ⅰ)函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702139716.png)
是否关于1可线性分解,关键是看是否存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702263324.png)
使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702279871.png)
成立,若成立,是关于1可线性分解,否则不是关于1可线性分解,故看
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702295820.png)
是否有解,构造函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702310926.png)
,看它是否有零点,而
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702326924.png)
,观察得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702341646.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702357676.png)
,有根的存在性定理可得存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702373636.png)
,使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702279871.png)
;(Ⅱ)先确定定义域为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702404553.png)
,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702154755.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702170523.png)
关于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702185283.png)
可线性分解,即存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702466453.png)
,使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702482838.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240247024971338.png)
有解,整理得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702513595.png)
有解,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702529704.png)
,从而求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702185283.png)
的取值范围;(Ⅲ)证明不等式:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240247022171021.png)
,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702575337.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702591691.png)
,对
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702607469.png)
求导,判断最大值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702622453.png)
,可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702638459.png)
,分别令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702653641.png)
,叠加可得证结论.
试题解析:(Ⅰ)函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702139716.png)
的定义域是R,若是关于1可线性分解,
则定义域内存在实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702107321.png)
,使得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702279871.png)
.
构造函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702310926.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702731886.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702763706.png)
.
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702794557.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702809567.png)
且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702825507.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702841317.png)
上是连续的,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702825507.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702872459.png)
上至少存在一个零点.
即存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702373636.png)
,使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702279871.png)
. 4分
(Ⅱ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702607469.png)
的定义域为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702404553.png)
.
由已知,存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702466453.png)
,使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702482838.png)
.
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240247029971354.png)
.
整理,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703012860.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703028874.png)
.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702513595.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703059595.png)
.
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703075703.png)
且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703090395.png)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703106447.png)
.
∴a的取值范围是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702232713.png)
. 9分
(Ⅲ)由(Ⅱ)知,a =1,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703153693.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703168752.png)
.
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703184584.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703199556.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702607469.png)
的单调递增区间是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702872459.png)
,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703246650.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703262561.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702607469.png)
的单调递减区间是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703309543.png)
,因此
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703324673.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702607469.png)
的最大值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702622453.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703371746.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703371733.png)
,因此得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703387392.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703402434.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703418480.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703433182.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024703433495.png)
,以上各式相加得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240247034491049.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240247034801671.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240247034801145.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240247022171021.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024702232632.png)
. 14分