本试题主要是考查了导数在研究函数中的运用
(1)利用函数在两个点处取得极值,可知极值点处导数为零得到参数的值。
(2)结合根与系数的关系和参数a,b的值,得到了函数关系式,那么要是等式成立,利用导数得到b的最值。
解:(1)
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333603551.png)
是函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333463447.png)
的两个极值点,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333634521.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333650547.png)
。∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333681657.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333697701.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333712563.png)
。∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333525839.png)
。……4分
(2)∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333837422.png)
是函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333463447.png)
的两个极值点,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333884754.png)
。
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333837422.png)
是方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333931789.png)
的两根。
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333946724.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333993426.png)
对一切
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334009617.png)
恒成立。
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334040774.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334055631.png)
,……6分
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334071399.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334102508.png)
。
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232303341182130.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334133637.png)
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232303341491008.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334180696.png)
。……8分
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334196451.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334211709.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334227499.png)
。令
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334258777.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334274830.png)
。
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334289472.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334305588.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334336491.png)
在(0,4)内是增函数;
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334352499.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334383585.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334336491.png)
在(4,6)内是减函数。……10分
∴当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334414372.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334336491.png)
有极大值为96,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334336491.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230334492440.png)
上的最大值是96,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333509299.png)
的最大值是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823230333556415.png)
。……12分