【题文】(本小题满分12分)二次函数满足,且最小值是.(1)求的解析式;(2)实数,函数,若在区间上单调递减,求实数的取值范围.
【题文】(本小题满分12分)二次函数满足,且最小值是.(1)求的解析式;(2)实数,函数,若在区间上单调递减,求实数的取值范围.
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【题文】(本小题满分12分)二次函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190134-24388.png)
满足
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190135-56679.png)
,且最小值是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190135-29442.png)
.
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190134-24388.png)
的解析式;
(2)实数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190135-93056.png)
,函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190135-26947.png)
,若
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-14710.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-23453.png)
上单调递减,求实数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190137-23524.png)
的取值范围.
答案
【答案】(1)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190137-77279.png)
;(2)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190137-49085.png)
.
解析
【解析】
试题分析:(1)根据条件
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190135-56679.png)
可设
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190137-91438.png)
,配方可得
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190137-62189.png)
,再由
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190134-24388.png)
的最小值是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190135-29442.png)
,从而
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190138-47026.png)
,即有
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190138-87968.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190137-77279.png)
;(2)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190138-95487.png)
,
从而
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190139-94080.png)
,因此
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-14710.png)
存在两个极值点
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190139-68976.png)
或
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190139-40412.png)
,再由条件
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-14710.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-23453.png)
上单调递减,因此需对
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190139-24234.png)
和
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190139-15712.png)
的大小关系进行分类讨论,即可得到关于
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190137-23524.png)
的不等式组, 当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-28331.png)
,即
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-12102.png)
时,由
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-90586.png)
,得
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-18504.png)
, ∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-14710.png)
的减区间是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-59803.png)
,又∵
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-14710.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-23453.png)
上单调递减,∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190141-19199.png)
(满足
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-12102.png)
),当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190141-89783.png)
,即
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190141-35923.png)
时,由
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-90586.png)
,得
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190141-31390.png)
,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-14710.png)
的减区间是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190142-60854.png)
,又∵
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-14710.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-23453.png)
上单调递减,∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190142-16739.png)
(满足
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190141-35923.png)
),即实数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190137-23524.png)
的取值范围为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190137-49085.png)
.
试题解析:(1)由二次函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190134-24388.png)
满足
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190135-56679.png)
,设
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190137-91438.png)
, 2分
则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190142-45778.png)
,又∵
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190134-24388.png)
的最小值是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190135-29442.png)
,故
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190138-47026.png)
,解得
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190138-87968.png)
,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190137-77279.png)
; 6分 ;
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190142-17533.png)
, 7分
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190143-36181.png)
,由
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190143-38834.png)
,得
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190139-68976.png)
或
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190139-40412.png)
,又∵
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190135-93056.png)
,故
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190143-34481.png)
, 8分 当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-28331.png)
,即
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-12102.png)
时,由
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-90586.png)
,得
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-18504.png)
,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-14710.png)
的减区间是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-59803.png)
,又∵
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-14710.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-23453.png)
上单调递减,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190141-19199.png)
(满足
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-12102.png)
), 10分
当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190141-89783.png)
,即
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190141-35923.png)
时,由
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190140-90586.png)
,得
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190141-31390.png)
,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-14710.png)
的减区间是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190142-60854.png)
,又∵
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-14710.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190136-23453.png)
上单调递减,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190142-16739.png)
(满足
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190141-35923.png)
),综上所述得
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190143-40892.png)
,或
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190144-52288.png)
,
∴实数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190137-23524.png)
的取值范围为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190137-49085.png)
. 12分 .
考点:1.二次函数的解析式;2.导数的运用.
举一反三
【题文】已知x, y,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190125-79419.png)
R,且
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190125-27484.png)
,则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190126-37937.png)
的最小值是( )
【题文】下列命题为真命题的是
.(用序号表示即可)
①cos1>cos2>cos3;
②若
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190112-98419.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190112-88287.png)
且
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190112-98419.png)
=n+3(n=1、2、3),则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190113-52037.png)
;
③若
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190113-11610.png)
、
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190113-58576.png)
、
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190114-78217.png)
分别为双曲线
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190114-37334.png)
=1、
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190115-29135.png)
=1、
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190116-99990.png)
=1的离心率,则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190113-11610.png)
>
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190113-58576.png)
>
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190114-78217.png)
;
④若
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190116-13596.png)
,则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190116-20339.png)
【题文】若二次函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190045-71988.png)
,满足
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190045-68523.png)
且
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190045-99277.png)
=2.
(Ⅰ)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190045-85200.png)
的解析式;
(Ⅱ)若存在
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190046-47248.png)
,使不等式
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190046-13267.png)
成立,求实数m的取值范围.
【题文】(本小题满分12分)已知函数
(1)用单调性的定义判断函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190005-64995.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190006-64987.png)
上的单调性并加以证明;
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190005-64995.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190006-96123.png)
的最小值为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190007-70902.png)
,求
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325190007-48694.png)
的解析式.
【题文】下列函数中,在
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325185952-48241.png)
上单调递增的偶函数是( )
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