试题分析:(1)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060014925388.png)
时,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015486447.png)
是一个具体的三次函数,只须求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015486447.png)
的导函数,并令它为零求得其根;然后列出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015533266.png)
的取值范围与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015549479.png)
的符号及
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015486447.png)
单调性的变化情况表,由此表可求得函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060014940429.png)
的极大值;(2)函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060014940429.png)
的图象与函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015018639.png)
的图象有三个不同的交点,等价于方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015658578.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015673719.png)
有三个不同的实数根,也等价于方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015689625.png)
有三个不同的实数根,从而可转化为直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015705416.png)
与函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240600157511014.png)
有三个不同的交点,画草图可知必须且只需:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015767867.png)
,所以利用导数求出函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240600157511014.png)
的极小值和极大值即可;(3)注意到函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015127602.png)
的图象与函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060014940429.png)
的图象之间的关系:将函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060014940429.png)
在x轴上方的图象不变,而将x轴下方的图象沿x轴翻折到x轴上方即得函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015127602.png)
的图象,由此可知要求函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015159426.png)
的单调减区间,只须先求出函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060014940429.png)
的单调区间,并求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060014940429.png)
的所有零点,结合图象就可写出函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015159426.png)
的单调减区间;注意分类讨论.
试题解析:(1)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015970383.png)
时,由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240600159851021.png)
=0,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016001358.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016017334.png)
, 2分
列表如下:
所以当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016017334.png)
时,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015486447.png)
取得极大值为5. 4分
(2)由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015658578.png)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015673719.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015689625.png)
, 6分
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016235734.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240600162821090.png)
,
列表,得
8分
由题意知,方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016469515.png)
有三个不同的根,故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015112283.png)
的取值范围是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015174703.png)
. 10分
(3)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240600165311086.png)
,
所以当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015221416.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060014940429.png)
在R上单调递增;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016594451.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016609518.png)
的两根为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016625524.png)
,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016656831.png)
,
所以此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060014940429.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016687714.png)
上递增,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015330773.png)
上递减,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016734717.png)
上递增;12分
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016750497.png)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016781355.png)
,或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016797578.png)
(*),
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016812492.png)
时,方程(*)无实根或有相等实根;当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015361553.png)
时,方程(*)有两根
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060016859681.png)
, 13分
从而
①当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015221416.png)
时,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015252442.png)
的单调减区间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015268517.png)
; 14分
②当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015283553.png)
时,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015252442.png)
的单调减区间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015268517.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015330773.png)
; 15分
③当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015361553.png)
时,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015252442.png)
的单调减区间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015268517.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015424966.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060015455961.png)
. 16分