题目内容
(本题满分12分)
已知二次函数
的图象过点
,且与
轴有唯一的交点
.(1)求
的表达式;
(2)当
时,求函数
的最小值。
已知二次函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235944878948.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235944909429.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235944925266.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235944941467.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235944972447.png)
(2)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945003570.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235944972447.png)
(1)
(2) 当
时,最小值为
,
当
时,最小值为
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945050691.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945081522.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945112705.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945128404.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945284495.png)
试题分析:(1)依题意得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945299305.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945346567.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945362599.png)
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945393337.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945424403.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945299305.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945050691.png)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945487839.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945502332.png)
结合图象可知,当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945081522.png)
所以最小值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945112705.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945128404.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945596407.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945627418.png)
所以最小值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823235945284495.png)
点评:求闭区间上二次函数的值域时,要结合函数的图象进行求解,不要出现简单的把端点代入求解的错误.
![](http://thumb2018.1010pic.com/images/loading.gif)
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