题目内容
已知二次函数
的图象与x轴有两个不同的公共点,且
,当
时,恒有
.
(1)当
时,求不等式
的解集;
(2)若以二次函数的图象与坐标轴的三个交点为顶点的三角形的面积为8,且
,求a的值;
(3)若
,且
对所有
恒成立,求正实数m的最小值.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605193699.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605224299.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605240261.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605271307.gif)
(1)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605287437.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605302310.gif)
(2)若以二次函数的图象与坐标轴的三个交点为顶点的三角形的面积为8,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605318283.gif)
(3)若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605334302.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605349550.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605365299.gif)
(1)当
,c=2时,
,f(x)的图像与x轴有两个不同交点,
因为
,设另一个根为x1,则2x1=6,x1=3. …………2分
则
的解集为
. …………4分
(2) 函数f(x)的图像与x轴有两个交点,因
,
设另一个根为
,则
于是
. …………6分
又当
时,恒有
,则
,则三交点为
,8分
这三交点为顶点的三角形的面积为
,且
,
解得
. ………10分
(3)当
时,恒有
,则
,
所以f(x)在
上是单调递减的,且在
处取到最大值1, ………12分
要使
,对所有
恒成立,
必须
成立,
,
解得
或
, 而
,
所以m的最小值为2. ………16分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605380266.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605396583.gif)
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605412301.gif)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605302310.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605458485.gif)
(2) 函数f(x)的图像与x轴有两个交点,因
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605224299.gif)
设另一个根为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605490198.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605505404.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605521378.gif)
又当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605240261.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605271307.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605583265.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605599660.gif)
这三交点为顶点的三角形的面积为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605614603.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605630277.gif)
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605646338.gif)
(3)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605240261.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605271307.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605583265.gif)
所以f(x)在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605724261.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605755228.gif)
要使
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605770546.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605786304.gif)
必须
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605802610.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605833758.gif)
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605848355.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605864361.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823190605895261.gif)
所以m的最小值为2. ………16分
略
![](http://thumb2018.1010pic.com/images/loading.gif)
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