题目
题型:解答题难度:简单来源:不详
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201849-15912.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-57257.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-38391.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201849-15912.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-16405.gif)
(Ⅰ)判断下面两个函数是否具有性质
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-16405.gif)
①
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-30959.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-49123.gif)
(Ⅱ)若函数
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201849-15912.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-16405.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-46466.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-98169.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-54013.gif)
求证:对任意
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-29891.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-12571.gif)
(Ⅲ)在(Ⅱ)的条件下,是否对任意
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-81974.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-85818.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-87839.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-16405.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-29804.gif)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-26786.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-45031.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-38391.gif)
此函数为具有性质
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-16405.gif)
②函数
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-90203.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-16405.gif)
例如,当
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-27948.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-69830.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-12364.gif)
所以,
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-66141.gif)
此函数不具有性质
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-16405.gif)
(Ⅱ)假设
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-33481.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-79809.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-77809.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-83657.gif)
因为函数
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201849-15912.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-16405.gif)
所以,对于任意
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-54013.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-53660.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-88861.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-20087.gif)
与
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-40935.gif)
所以,对任意的
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-29891.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-77613.gif)
(Ⅲ)不成立.
例如
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-25999.gif)
证明:当
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-88192.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201852-16004.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201852-32404.gif)
当
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-88192.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201852-16004.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201852-19151.gif)
所以,函数
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201849-15912.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-57257.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-38391.gif)
即函数
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201849-15912.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-16405.gif)
而当
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201852-55127.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201852-68870.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201851-88192.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201852-17321.gif)
所以,在(Ⅱ)的条件下,“对任意
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-81974.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201850-85818.gif)
(其他反例仿此给分.
如
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201852-99140.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201852-88762.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201852-19892.gif)
解析
核心考点
试题【若函数对任意的,均有,则称函数具有性质.(Ⅰ)判断下面两个函数是否具有性质,并说明理由.①; ②.(Ⅱ)若函数具有性质,且(),求证:对任意有;(Ⅲ)在(Ⅱ】;主要考察你对指数函数图象及性质等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201843-72110.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201843-79553.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201843-86837.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201843-29602.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201844-48119.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201844-10261.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201844-48119.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201844-85109.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201844-48119.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201844-84808.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201845-83537.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201841-91209.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201841-39015.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201841-44495.png)
A.![]() |
B.![]() |
C.![]() |
D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201839-96662.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201839-74504.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201839-50016.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201839-82345.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201839-64311.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201840-12283.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201840-96294.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201840-77801.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201840-45051.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201840-84150.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201840-11952.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201840-96294.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201840-89203.png)
A.![]() |
B.![]() |
C.![]() |
D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201836-24324.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201836-10697.png)
A.1 | B.2 | C.3 | D.4 |
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201835-25577.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201835-20215.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201835-89664.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201835-95341.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201835-34193.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201835-61691.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819201835-61624.png)
A.2 | B.3 | C.4 | D.6 |
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