题目
题型:解答题难度:简单来源:不详
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192755-51930.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192755-12574.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-81233.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-12109.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-41282.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192755-51930.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-42398.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-12109.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-11616.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192755-51930.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-42398.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-96721.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-32501.png)
(1)若
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-10191.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-86653.png)
(2)对任给的“2阶不减函数”
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192755-51930.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-45038.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-51220.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192755-51930.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-10949.png)
(2)所有满足题设的
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-97190.png)
解析
试题分析:解:(1)依题意,
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-17757.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-34688.png)
故
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-53583.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-80157.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-43883.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-10949.png)
而当
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-10949.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-92426.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-34688.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-10949.png)
(2)①先证
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-24966.png)
若不存在正实数
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-53550.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-77930.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-51781.png)
假设存在正实数
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-53550.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-77930.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-28653.png)
由题意,当
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-43883.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-34491.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-52801.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-34688.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-46036.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-90709.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-78219.png)
故必存在
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-87967.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-93283.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-59452.png)
这与
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-90077.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-97190.png)
所以当
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-43883.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-51781.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-24966.png)
②再证
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-34566.png)
假设存在正实数
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-91851.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-91693.png)
则对于任意
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-31288.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192758-24352.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192758-91884.png)
这与①矛盾,故假设不成立,
所以
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192757-34566.png)
综上得
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192758-17941.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192758-59390.png)
故所有满足题设的
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192756-97190.png)
点评:主要是考查了新定义的运用,以及函数与方程的运用,属于中档题。
核心考点
试题【设是定义在的可导函数,且不恒为0,记.若对定义域内的每一个,总有,则称为“阶负函数 ”;若对定义域内的每一个,总有,则称为“阶不减函数”(为函数的导函数).(1】;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/20190819/20190819192739-41475.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192739-85249.png)
A.![]() | B.1 | C.4 | D.![]() |
A.![]() | B.![]() |
C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192732-36277.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192732-63472.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192732-23639.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192732-31041.png)
(1)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192733-87405.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192733-11088.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192733-98602.png)
(3)证明不等式
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192733-68804.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192733-21059.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192730-11783.png)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192730-17389.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192730-30237.png)
(2)若对任意
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192730-10943.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192730-24263.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192730-30237.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192729-67608.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190819/20190819192729-83186.png)
A.5 | B.6 | C.7 | D.8 |
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