试题分析:(I) 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647429545.png)
时,试讨论
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647444447.png)
的单调性,首先确定定义域
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647897648.png)
,可通过单调性的定义,或求导确定单调性,由于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240306479121179.png)
,含有对数函数,可通过求导来确定单调区间,对函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647928447.png)
求导得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240306479431010.png)
,由此需对参数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647943283.png)
讨论,分
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647553365.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647663440.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647709531.png)
三种情况,判断导数的符号,从而得单调性;(II)设
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647460748.png)
,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647475429.png)
时,若对任意
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647491564.png)
,存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647507571.png)
,使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647522704.png)
,求实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647538284.png)
取值范围,由题意可知,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647475429.png)
时,若对任意
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647491564.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647444447.png)
的最小值大于或等于当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647507571.png)
时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648146442.png)
的最小值即可,由(I)知,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647475429.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647444447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647569383.png)
单调递减,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648193417.png)
单调递增.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648209872.png)
,只需求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648146442.png)
的最小值,由于本题属于对称轴不确定,需讨论,从而确定实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647538284.png)
取值范围.也可用分离参数法来求.
试题解析:(I)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648255887.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240306482711038.png)
(
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648287385.png)
) 3分
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648302238.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647553365.png)
时,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647569383.png)
上,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647585573.png)
,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647600510.png)
上,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647616576.png)
,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647444447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647569383.png)
上单调递减,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647600510.png)
上单调递增; 4分
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648411303.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647663440.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648458576.png)
,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647444447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647694525.png)
单调递减; 5分
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648505293.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647709531.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648536462.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647725479.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647585573.png)
,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647444447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647569383.png)
上单调递减;
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647772622.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647616576.png)
,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647444447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647787518.png)
上单调递增;
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647803744.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647585573.png)
,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647444447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647834636.png)
上单调递减. 7分
(II)若对任意
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647491564.png)
,存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647507571.png)
,使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647522704.png)
成立,只需
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648755741.png)
9分
由(I)知,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647475429.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647444447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647569383.png)
单调递减,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648193417.png)
单调递增.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648209872.png)
, 11分
法一:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647460748.png)
,对称轴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648848487.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648302238.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648879494.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648895396.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648911919.png)
,得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648926581.png)
;
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648411303.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648957499.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648973395.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648973891.png)
,得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648973395.png)
;
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030648505293.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030649035557.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030649051474.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030649067994.png)
,得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030649051474.png)
. 14分
综上:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647865505.png)
. 15分
法二:
参变量分离:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030649098630.png)
, 13分
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030649113739.png)
,只需
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030649129685.png)
,可知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030649145468.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030649160402.png)
上单调递增,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030649207922.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824030647865505.png)
. 15分