题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222137-32410.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222137-84467.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222138-48900.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222138-46377.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222138-54714.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222139-32864.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222139-59692.png)
(1)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222139-91534.png)
(2)证明:对任意的
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222139-93514.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222140-44742.png)
(3)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222140-73797.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222137-32410.png)
(4)若
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222140-78352.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222140-34050.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222140-75787.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222141-18551.png)
(3)利用单调性定义即可证明(4)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222141-27362.png)
解析
试题分析:(1)证明:令
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222140-75787.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222141-22919.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222138-48900.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222139-91534.png)
(2)证明:由已知当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222138-46377.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222138-54714.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222139-91534.png)
故当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222141-46473.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222140-44742.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222142-28954.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222142-86090.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222142-29797.png)
而
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222142-67500.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222143-19713.png)
可得
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222143-43643.png)
综上:对任意的
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222139-93514.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222140-44742.png)
(3)证明:设
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222143-35621.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222143-16103.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222143-93711.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222144-83055.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222144-59557.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222144-75369.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222144-83905.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222145-81967.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222145-59837.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222137-32410.png)
(4)由
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222140-78352.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222145-32805.png)
又因为
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222145-28344.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222137-32410.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222146-14552.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222146-55670.png)
所以:所求
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222140-34050.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222141-27362.png)
点评:求解抽象函数问题,主要的方法是赋值法,证明抽象函数的单调性只能用定义,证明时要尽量化简到最简单.
核心考点
试题【(12分)定义在上的函数,,当时,.且对任意的有。(1)证明:;(2)证明:对任意的,恒有;(3)证明:是上的增函数;(4)若,求的取值范围。】;主要考察你对函数的相关概念等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222106-24401.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204222107-48956.png)
A.1 | B.2 | C.4 | D.8 |
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221952-79430.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221952-56461.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221952-79430.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221952-39348.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221953-10310.png)
万元;投资
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221952-56461.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221953-16477.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221953-12390.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221954-97845.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221954-63613.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221954-43380.png)
万元来投资这两个项目,则分别投资多少万元能获得最大利润?最大利润是多少?
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221942-23548.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221943-82259.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221943-69226.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221935-57672.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221935-22392.png)
使得
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221935-53909.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204221936-99023.png)
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