题目内容
(本题满分15分)本题文科做.
已知二次函数
的二次项系数为
,且不等式
的解集为
.
(1)若方程
有两个相等的实数根, 求
的解析式;
(2)若
的最大值为正数,求
的取值范围.
已知二次函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116273463.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116382278.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116398604.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116414447.png)
(1)若方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116445702.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116273463.png)
(2)若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116273463.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116382278.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116710886.png)
第一问利用![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201168821107.png)
∴
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201169751493.png)
然后由方程
因为方程②有两个相等的根,所以
得到。
第二问,由![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201170531888.png)
及
由
解得 ![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116710886.png)
解:(1)![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201168821107.png)
∴
所以
…………………………2分
①
由方程
② ……………………4分
因为方程②有两个相等的根,所以
,
即
………………………6分
由于
代入①得
的解析式为
……………………………8分
(若本题没有舍去“
”第一小问得6分)
(2)由![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201174431891.png)
及
……………………………12分
由
解得 ![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116710886.png)
故当
的最大值为正数时,实数a的取值范围是
…15分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201168821107.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201168971000.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116944387.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201169751493.png)
然后由方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201170061281.png)
因为方程②有两个相等的根,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220117022944.png)
第二问,由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201170531888.png)
及
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201170691428.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201170841030.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116710886.png)
解:(1)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201168821107.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201168971000.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116944387.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201169751493.png)
由方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201170061281.png)
因为方程②有两个相等的根,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220117022944.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201173501175.png)
由于
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220117365855.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220117381447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220117412951.png)
(若本题没有舍去“
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220117428336.png)
(2)由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201174431891.png)
及
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201170691428.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232201170841030.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220116710886.png)
故当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220117381447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823220117786987.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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