【题文】(本小题满分12分) 已知函数满足,对任意,都有,且.(Ⅰ)求函数的解析式;(Ⅱ)若,使方程成立,求实数的取值范围.
题型:难度:来源:
【题文】(本小题满分12分) 已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063416-10738.png)
满足
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063417-88792.png)
,对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063417-92603.png)
,都有
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063417-73450.png)
,且
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063418-32512.png)
.
(Ⅰ)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063418-79434.png)
的解析式;
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063419-26947.png)
,使方程
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063419-50054.png)
成立,求实数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063419-12013.png)
的取值范围.
答案
【答案】(Ⅰ)
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063420-74733.png)
;(Ⅱ)
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063420-91832.png)
.
解析
【解析】
试题分析: (Ⅰ)因为
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063416-10738.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063417-88792.png)
,所以
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063421-56597.png)
,∵对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063417-92603.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063418-32512.png)
,∴
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063421-99867.png)
的对称轴为直线
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063421-18624.png)
,求得
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063421-91758.png)
;又因为对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063417-92603.png)
都有
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063417-73450.png)
,利用函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063418-79434.png)
的图象结合判别式,求得
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063422-83876.png)
,所以
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063420-74733.png)
;(Ⅱ)由
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063419-50054.png)
得
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063422-19298.png)
,∴方程
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063422-19298.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063423-42895.png)
有解,则
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063423-75196.png)
在函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063423-53912.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063423-42895.png)
值域内,求出
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063423-53912.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063423-42895.png)
的值域,使
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063423-75196.png)
在函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063424-29357.png)
的值域内,求解即可.
试题解析:(Ⅰ)∵
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063416-10738.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063417-88792.png)
,∴
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063421-56597.png)
1分
又∵对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063417-92603.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063418-32512.png)
,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063421-99867.png)
图象的对称轴为直线
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063421-18624.png)
,则
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063424-12530.png)
,∴
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063421-91758.png)
2分
又∵对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063417-92603.png)
都有
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063417-73450.png)
,即
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063424-10734.png)
对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063417-92603.png)
都成立,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063425-63268.png)
, 4分
故
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063422-83876.png)
,∴
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063420-74733.png)
6分
(Ⅱ)由
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063419-50054.png)
得
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063422-19298.png)
,由题意知方程
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063422-19298.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063423-42895.png)
有解.令
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063425-57575.png)
,∴
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063425-72288.png)
8分
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063426-93540.png)
,∴
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063426-90646.png)
, 11分
所以满足题意的实数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063419-12013.png)
取值范围
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063420-91832.png)
. 12分
考点:①求二次函数的解析式;②利用一元二次方程有解求参数范围.
举一反三
【题文】(12分)已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063345-48415.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063346-30206.png)
上是减函数,在
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063346-93372.png)
上是增函数,且对应方程两个实根
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063346-70449.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063346-84914.png)
满足
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063346-31413.png)
,
(1)求二次函数的解析式;
(2)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063347-28687.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063347-68428.png)
上的值域
【题文】(12分)已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063335-18275.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063336-24435.png)
上是减函数,在
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063336-32592.png)
上是增函数,且对应方程两个实根
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063337-98439.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063337-11220.png)
满足
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063337-23375.png)
,
(1)求二次函数的解析式;
(2)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063338-46123.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063338-63229.png)
上的值域
【题文】(本题满分12分)若二次函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063326-34362.png)
,满足
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063327-77030.png)
且
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063327-41793.png)
=2.
(1)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063327-61617.png)
的解析式;
(2)若存在
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063327-72203.png)
,使不等式
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063327-23573.png)
成立,求实数m的取值范围.
【题文】(本题满分12分)若二次函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063313-72467.png)
,满足
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063313-73849.png)
且
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063314-67360.png)
=2.
(1)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063314-64313.png)
的解析式;
(2)若存在
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063314-48274.png)
,使不等式
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063314-85051.png)
成立,求实数m的取值范围.
【题文】(12分)(原创)已知二次函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063253-79059.png)
满足以下要求:
①函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063254-90163.png)
的值域为
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063254-86784.png)
;②
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063254-90782.png)
对
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063255-47684.png)
恒成立。
(1)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063254-90163.png)
的解析式;
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063255-62009.png)
,求
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063255-46741.png)
时
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327063255-97055.png)
的值域。
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