【错解分析】利用一阶导数求函数的极大值和极小值的方法是导数在研究函数性质方面的继续深入 是导数应用的关键知识点,通过对函数极值的判定,可使学生加深对函数单调性与其导数关系的理解.
【正解】
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240015344521130.png)
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534483465.png)
=0得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534499869.png)
.
(1)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240015345301315.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534545283.png)
<0或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534545283.png)
>4时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534499869.png)
有两个不同的实根
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534608300.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534623331.png)
,
不妨设
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534608300.png)
<
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534623331.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534670884.png)
,
易判断
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534686503.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534701295.png)
和
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534717330.png)
两侧的符号都相反,即此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534733447.png)
有两个极值点.
(2)当△=0即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534545283.png)
=0或
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534545283.png)
=4时,方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534499869.png)
有两个相同的实根
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534811433.png)
,于是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534826783.png)
,故在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534701295.png)
的两侧均有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534483465.png)
>0,因此
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534733447.png)
无极值.
(3)当△<0即0<
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534545283.png)
<4时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534499869.png)
无实数根,
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240015349511225.png)
,
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534733447.png)
为增函数,此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534733447.png)
无极值.
综上所述:当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240015350131654.png)
无极值点.
【点评】此题考查的是可导函数在某点取得极值的充要条件,即:设
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001535029559.png)
在某个区间内可导,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001534733447.png)
在某点取得极值的充要条件是该点的导数为零且在该点两侧的导数值异号.本题从逆向思维的角度出发,根据题设结构进行逆向联想,合理地实现了问题的转化,使抽象的问题具体化