试题分析:(1)依题意,计算
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118261820.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118276677.png)
.
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060117683587.png)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118323323.png)
,再据
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118339516.png)
,可得“拐点”坐标是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118120459.png)
.
(2)由(1)知“拐点”坐标是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118120459.png)
.
根据定义(2),考查
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118401648.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118432827.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118448873.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118463777.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118479497.png)
,
作出结论:
一般地,三次函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118151971.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060117621491.png)
的“拐点”是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118183994.png)
,它就是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060117949447.png)
的对称中心.
或者:任何一个三次函数都有拐点;任何一个三次函数都有对称中心;任何一个三次函数平移后可以是奇函数.
(3)根据(2)写出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240601185881170.png)
或写出一个具体的函数,如
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118229856.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118245764.png)
.
试题解析:(1)依题意,得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118261820.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118666688.png)
。 2分
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060117683587.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118713484.png)
。∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118323323.png)
,又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118339516.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060117933813.png)
的“拐点”坐标是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118120459.png)
.。 4分
(2)由(1)知“拐点”坐标是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118120459.png)
.
而
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118401648.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118432827.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118448873.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118463777.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118479497.png)
,
由定义(2)知:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118978870.png)
关于点
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118120459.png)
对称。 8分
一般地,三次函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118151971.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060117621491.png)
的“拐点”是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118183994.png)
,它就是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060117949447.png)
的对称中心. 10分
(或者:任何一个三次函数都有拐点;任何一个三次函数都有对称中心;任何一个三次函数平移后可以是奇函数 )都可以给分
(3)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240601191031183.png)
或写出一个具体的函数,如
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118229856.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824060118245764.png)
. 12分