题目内容
设函数
,曲线
在点
处的切线方程为![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043041359.png)
(1)确定
的值
(2)若过点(0,2)可做曲线
的三条不同切线,求
的取值范围
(3)设曲线
在点
处的切线都过点(0,2),证明:当
时,![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043134751.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240200429941277.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043025561.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043025582.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043041359.png)
(1)确定
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043056370.png)
(2)若过点(0,2)可做曲线
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043072447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043087283.png)
(3)设曲线
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043025561.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043119776.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043119436.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043134751.png)
(1)![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043150489.png)
(2)![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043165577.png)
(3)运用反证法来加以证明即可。
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043150489.png)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043165577.png)
(3)运用反证法来加以证明即可。
试题分析:(1)根据题意,由于函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240200429941277.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043025561.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043025582.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043041359.png)
则可知f’(0)=0,得到
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043150489.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043275955.png)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043290690.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240200433061056.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240200433211393.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043337480.png)
所以,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240200433681366.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043384925.png)
设
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240200433991975.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240200434151590.png)
(3)反证法:由题知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043415901.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043431911.png)
两式作差得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240200434461254.png)
若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043493908.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824020043509260.png)
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240200435241268.png)
与已知矛盾
点评:主要是考查了导数的几何意义以及函数的最值问题,属于中档题。
![](http://thumb2018.1010pic.com/images/loading.gif)
练习册系列答案
相关题目