由在点
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220319982472.png)
处的切线方程是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220319997498.png)
可得出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320434526.png)
,k=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320450456.png)
=0;
列式求解;
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320200782.png)
恒成立,则即最高点与最低点纵标差
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320496333.png)
即可,转化为求函数在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320106432.png)
上的
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320715776.png)
问题
解:(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320777235.png)
函数
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220319857447.png)
过点
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220319872485.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320840760.png)
------------1分
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320855975.png)
,函数
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220319857447.png)
在点
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220319982472.png)
处的切线方程是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220319997498.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320980909.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232203209961046.png)
-----------------------3分
解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320309645.png)
,故
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320340670.png)
--------------------5分
(2)由(1)知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220321042683.png)
,令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220321058560.png)
解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220321089342.png)
,-------------6分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232203211201062.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220321136195.png)
在区间
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320106432.png)
上
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232203211831020.png)
,-----------------8分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220321136195.png)
对于区间
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320106432.png)
上任意两个自变量的值
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220320122420.png)
,
都有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232203212761321.png)
,---------------------9分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823220321308497.png)
,所以t的最小值是20