【题文】已知二次函数,(1)若写出函数的单调增区间和减区间(2)若求函数的最大值和最小值:(3)若函数在上是单调函数,求实数的取值范围.
题型:难度:来源:
【题文】已知二次函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214938-45836.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214938-46020.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214938-56180.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214939-17109.png)
(1)若
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214939-79753.png)
写出函数的单调增区间和减区间
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214939-92409.png)
求函数的最大值和最小值:
(3)若函数在
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214939-17109.png)
上是单调函数,求实数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214940-51595.png)
的取值范围.
答案
【答案】(1)单调递增区间为:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214940-91340.png)
,单调递减区间为:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214940-44054.png)
;(2)最大值为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214940-42525.png)
,最小值为:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214941-14503.png)
;(3)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214942-65994.png)
.
解析
【解析】
试题分析:(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214939-79753.png)
时,求出函数的对称轴,可得函数的单调区间;
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214939-92409.png)
时,求出函数的对称轴,利用函数在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214939-17109.png)
上的单调性,确定函数的最大值和最小值;
(3)求出函数的对称轴,利用函数在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214939-17109.png)
上是单调增函数,确定对称轴和区间之间的关系,求出实数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214940-51595.png)
的取值范围.
试题解析:(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214939-79753.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214942-51079.png)
,因为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214938-46020.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214938-56180.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214939-17109.png)
,所以函数的单调递增区间为:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214940-91340.png)
,单调递减区间为:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214940-44054.png)
.
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214939-92409.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214942-97969.png)
,因为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214938-46020.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214938-56180.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214939-17109.png)
,所以函数的单调递增区间为:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214943-16804.png)
,单调递减区间为:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214943-56368.png)
,所以函数的最大值为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214940-42525.png)
,最小值为:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214941-14503.png)
.
(3)由
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214943-91716.png)
可得:函数的对称轴为:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214943-82697.png)
,因为函数在
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214939-17109.png)
上是单调函数,所以
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214942-65994.png)
.
考点:二次函数性质的综合应用.
举一反三
【题文】已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214731-62263.png)
且此函数图象过点(1,5).
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![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214731-29980.png)
奇偶性;
(3)判断函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214731-29980.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214731-64418.png)
上的单调性?并用定义证明你的结论.
【题文】设定义在R上的函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214428-51977.png)
,对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214428-20443.png)
有
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214429-15717.png)
,且当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214429-11102.png)
时,恒有
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214430-50724.png)
,
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214430-79344.png)
;
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![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214431-34260.png)
时 ,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214428-51977.png)
为单调递增函数.
【题文】设函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214400-96514.png)
.
(Ⅰ)求函数y=f(x)的最小值.
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![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214400-28009.png)
恒成立,求实数a的取值范围.
【题文】设
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214208-64382.png)
,若函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214209-68658.png)
为单调递增函数,且对任意实数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214209-77312.png)
,都有
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214210-21274.png)
(
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214211-21561.png)
是自然对数的底数),则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214211-95917.png)
( )
A.1 | B.![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214212-89396.png) | C.3 | D.![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214212-84131.png) |
【题文】已知命题
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213744-63342.png)
:函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213744-19318.png)
为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213744-78071.png)
上单调减函数,实数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213745-12120.png)
满足不等式
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213745-66597.png)
.命题
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213745-84545.png)
:当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213746-42487.png)
,函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213746-95160.png)
.若命题
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213744-63342.png)
是命题
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213745-84545.png)
的充分不必要条件,求实数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213747-94029.png)
的取值范围。
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