试题分析:(1)令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014420699.png)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014436423.png)
,①当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014311524.png)
时,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014170447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014498489.png)
上单调递减,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014685595.png)
上单调递增。此时最小值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014342566.png)
;②当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014358446.png)
时,函数在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014732476.png)
上单调递增,此时最小值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014373579.png)
。
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240110147791009.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014794533.png)
上有且仅有仅有一个根,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014810691.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014794533.png)
上有且仅有仅有一个根,令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014841839.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014872978.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240110148881080.png)
上递增,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014904870.png)
。
(3)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240110149191593.png)
,由题意知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014935732.png)
有两个不同的实数根
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014280617.png)
,等价于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014966617.png)
有两个不同的实数根
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014997422.png)
,等价于直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011015013416.png)
与函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011015028764.png)
的图像有两个不同的交点。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011015091725.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240110151061511.png)
所以当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240110151221091.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014997422.png)
存在,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011015153420.png)
的值随着
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011015169283.png)
的增大而增大。
而当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011015184582.png)
时,则有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240110152001156.png)
,两式相减得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240110152311225.png)
代入
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011015184582.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011015262865.png)
此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011015278883.png)
,所以实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011015169283.png)
的取值范围为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824011014404912.png)
点评:第一小题求最值需对参数分情况讨论从而确定最值点的位置,第二小题将方程的根的情况转化为函数最值得判定,这种转化方法包括将不等式恒成立问题转化为函数最值问题都是函数题目中经常用到的思路,须加以重视