题目内容
已知
为奇函数
的极大值点,
(1)求
的解析式;
(2)若
在曲线
上,过点
作该曲线的切线,求切线方程.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224412649333.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232244128211123.png)
(1)求
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224412836448.png)
(2)若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413070516.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413086562.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413257270.png)
(1)
;
(2) 切线方程为
或![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413320602.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413273661.png)
(2) 切线方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413304470.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413320602.png)
本试题主要是考查而来导数在研究函数中的运用,导数的几何意义的运用,导数的极值的运用。
(1)因为
为奇函数
的极大值点,可知参数a,b的值,得到解析式。
(2)由(1)知
,设切点为
,则切线方程为
.
点在切线上,有
解方程得到切线的坐标,进而得到方程。
解:(1)
为奇函数,故
.
.
分
,得
或
.
分
当
时,
为
的极小值点,与已知矛盾,舍去.
故
.
分
(2)由(1)知
,设切点为
,则切线方程为
.
点在切线上,有
,
,
,
,
即
.
或
,此时原曲线有两条切线.
分
切线方程为
或
.
分
(1)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224412649333.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232244128211123.png)
(2)由(1)知
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413382497.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413398669.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232244134291021.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413257270.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232244134761042.png)
解:(1)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224412836448.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413523388.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413569759.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413772309.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413788751.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413803379.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413835387.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413866267.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413835387.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224412649333.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224412836448.png)
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413273661.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413959318.png)
(2)由(1)知
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413382497.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413398669.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232244134291021.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413257270.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232244140371056.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232244140691041.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232244140841378.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224414100795.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224414115850.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224414131385.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224414162507.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224414178323.png)
切线方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413304470.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224413320602.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823224414240351.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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