试题分析:(Ⅰ)先写出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806239366.png)
时的函数解析式以及定义域:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240248063801053.png)
,对函数求导并且求得函数的零点,结合导数的正负判断函数在零点所分的各个区间上的单调性,从而得到函数的极值点,求得极值点对应的函数值即可;(Ⅱ)先求出函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240248063951066.png)
的导数,将问题“
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806270447.png)
在定义域内无极值”转化为“
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806426568.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806442572.png)
在定义域上恒成立”,那么设
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806458816.png)
分两种情况进行讨论,分别为方程无解时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806458409.png)
,以及方程有解时保证
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806473774.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240248064891156.png)
成立,解不等式及不等式组,求两种情况下解的并集.
试题解析:(Ⅰ)已知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806239366.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240248063801053.png)
, 1分
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240248065361090.png)
, 2分
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806551552.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806567336.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806567367.png)
. 3分
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806582650.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806598562.png)
;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806614436.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806629569.png)
. 4分
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806645901.png)
, 5分
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806270447.png)
取得极小值2,极大值
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806676496.png)
. 6分
(Ⅱ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240248063951066.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240248067071189.png)
, 7分
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806270447.png)
在定义域内无极值,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806426568.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806442572.png)
在定义域上恒成立. 9分
设
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806458816.png)
,根据图象可得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806458409.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240248064891156.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806348391.png)
. 11分
∴实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806302278.png)
的取值范围为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024806348391.png)
. 12分