试题分析:(1)当a=1,b=0时求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210192481.png)
,再把x=2代入即可求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000209849478.png)
的值;
(2)根据导数的几何意义可求
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210224593.png)
点(1,-11)在函数f(x)的图像上可建立关于a,b的两个方程,从而求出a,b的值.
(3)在(2)的条件下可求出f(x)的导数,利用
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210239691.png)
确定其单调增(减)区间即可.
解:1)求导数得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210255908.png)
,…………………………3分
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000209834519.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210302954.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000209958559.png)
…………………………………4分
(2)由于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000209865447.png)
的图像与直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210442596.png)
相切于点
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210458429.png)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210489852.png)
………………………6分
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240002105041006.png)
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000209974537.png)
……………………9分
(3)由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000209974537.png)
得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240002105671593.png)
……………10分
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210614569.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210629354.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210660406.png)
;由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210770562.png)
,
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210785464.png)
. --------------------13分
故函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000209865447.png)
在区间
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210005697.png)
上单调递增,在区间
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000210161467.png)
上单调递减. ---14分
点评:利用导数研究函数的单调区间,极值,最值是常考题型,要注意导数的几何意义是在某点处的切线的斜率,导数等于零的点不一定是极值点,要注意此点满足左正右负为极大值,此点处满足左负右正为极小值,两侧符号相同不是极值点.