题目内容
(本小题满分12分)
已知二次函数
,
,
的最小值为
.
⑴ 求函数
的解析式;
⑵ 设
,若
在
上是减函数,求实数
的取值范围;
已知二次函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232334315821159.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431613666.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431629444.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431660223.png)
⑴ 求函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431629444.png)
⑵ 设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431707823.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431738442.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431769320.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431878339.png)
⑴
. ⑵
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431894657.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431972539.png)
本试题主要是考查了二次函数的 解析式的求解,以及二次函数的最值的求解的综合运用。
(1)根据题意设
, ∵
的最小值为
,∴
,且
, ∴
,得到解析式。
(2)因为
,那么对属于参数m进行分类讨论,得到单调性,求解参数的范围。
解:⑴ 由题意设
,
∵
的最小值为
,
∴
,且
, ∴
,
.
⑵ ∵
,
① 当
时,
在[-1, 1]上是减函数,
符合题意.
② 当
时,对称轴方程为:
,
ⅰ)当
,即
时,二次函数的图象开口向上,
由
, 得
, ∴
;
ⅱ)当
, 即
时,二次函数的图象开口向下,
由
,得
, ∴
.
综上知,实数
的取值范围为
.
(1)根据题意设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431988725.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431629444.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431660223.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432237402.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432378478.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432393338.png)
(2)因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432409988.png)
解:⑴ 由题意设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431988725.png)
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431629444.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431660223.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432237402.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432378478.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432393338.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431894657.png)
⑵ ∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432409988.png)
① 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432596387.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432612605.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432596387.png)
② 当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432643416.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432674557.png)
ⅰ)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432690498.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432721403.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432768553.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432783510.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432799509.png)
ⅱ)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432814483.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432830423.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432846571.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432861527.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233432830423.png)
综上知,实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431878339.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233431972539.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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