【题文】已知函数,任取,定义集合,点满足,设,分别表示集合中元素的最大值和最小值,记,则(Ⅰ)函数的最大值为
【题文】已知函数,任取,定义集合,点满足,设,分别表示集合中元素的最大值和最小值,记,则(Ⅰ)函数的最大值为
题型:难度:来源:
【题文】已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050718-75957.png)
,任取
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050718-41555.png)
,定义集合
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050719-69971.png)
,点
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050719-19352.png)
满足
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050719-98845.png)
,设
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050719-58485.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050719-41814.png)
分别表示集合
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050720-89924.png)
中元素的最大值和最小值,记
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050720-75187.png)
,则
(Ⅰ)函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050720-70156.png)
的最大值为
;
(Ⅱ)函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050720-70156.png)
的单调区间为
.
答案
【答案】2;
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050721-85026.png)
.
解析
【解析】
试题分析:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050719-69971.png)
,点
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050721-94837.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050721-47911.png)
满足
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050719-98845.png)
表示以
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050721-55467.png)
点为圆心,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050722-63729.png)
为半径的圆及其内部及其内部函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050718-75957.png)
的图象上所有的点的纵坐标的集合,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050722-21412.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050722-51494.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050722-51085.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050723-20067.png)
,设
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050723-30186.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050723-56552.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050723-77020.png)
,则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050724-72049.png)
,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050724-52077.png)
,其中
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050724-34656.png)
是最高点
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050724-91110.png)
的横坐标,
同理
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050725-32228.png)
,其中
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050725-63729.png)
是最高点
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050724-91110.png)
的横坐标,
∴函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050725-16837.png)
的最大值是2(
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050725-13423.png)
或
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050726-34643.png)
时取得),单调增区间是
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050721-85026.png)
.
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050726-82794.png)
考点:函数的值域,抽象函数.
举一反三
【题文】若函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050626-67305.png)
是周期为5的奇函数,且满足
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050626-99856.png)
,则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050627-98483.png)
=
.
【题文】设函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050613-24042.png)
,则
.
【题文】已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325050559-11529.png)
在[-1,+ ∞)上是减函数,则a的取值范围是
.
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