题目
题型:解答题难度:简单来源:不详
已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004425-51049.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004425-70226.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004426-45620.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004426-30245.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004426-40744.png)
(Ⅰ)证明
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004426-88527.png)
(Ⅱ)证明
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004426-48923.png)
(Ⅲ)当(Ⅱ)中
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004427-57405.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004427-83757.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004427-71436.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004427-44050.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004427-13621.png)
(Ⅲ)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004428-55042.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004428-81547.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004428-57754.png)
解析
(1)对于任意的a>0,
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004428-19154.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004428-66944.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-31762.png)
(2) 令
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-44456.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-85917.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-62234.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-88469.png)
而
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-88580.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004430-42407.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004430-98511.png)
而
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004430-57032.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-88469.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004430-97973.png)
赋值法得到结论。
(3)由(Ⅱ)中的③知,当
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-62234.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004430-74637.png)
分析导数得到单调区间。
(Ⅰ)证明:对于任意的a>0,
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004428-19154.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004428-66944.png)
在①中取
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-31762.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004431-49263.png)
(Ⅱ)证法一:当
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004431-78318.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004431-27807.png)
取
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004431-90313.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004431-48544.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004432-63427.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004432-20558.png)
取
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004432-79289.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004432-51955.png)
综合②、③、④得
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004432-59481.png)
证法二:
令
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-44456.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-85917.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-62234.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-88469.png)
而
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-88580.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004430-42407.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004430-98511.png)
而
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004430-57032.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-88469.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004430-97973.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004433-36694.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-85917.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004433-35712.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004433-30918.png)
而
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004433-35712.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004433-70745.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004433-39334.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004434-75961.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004427-13621.png)
(Ⅲ)解法1:由(Ⅱ)中的③知,当
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-62234.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004430-74637.png)
从而
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004434-52293.png)
又因为k>0,由此可得
![]() | ![]() | ![]() | ![]() |
![]() | - | 0 | + |
![]() | ↘ | 极小值2 | ↗ |
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004428-55042.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004434-85115.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004435-95155.png)
解法2:由(Ⅱ)中的③知,当
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004429-62234.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004430-74637.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004435-71085.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004435-18042.png)
又因为k>0,所以
(i)当
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004436-83735.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004436-13492.png)
(ii)当
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004436-16554.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004428-55042.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004428-81547.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004428-57754.png)
核心考点
试题【(本小题满分14分)已知函数在上有定义,对任意实数和任意实数,都有. (Ⅰ)证明;(Ⅱ)证明(其中k和h均为常数);(Ⅲ)当(Ⅱ)中的时,设,讨论在内的单调性.】;主要考察你对分段函数等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004421-48195.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004421-84277.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004421-50594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004421-98781.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004421-21175.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004422-35150.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004422-61657.png)
A.0 | B.1 | C.2 | D.3 |
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004419-89288.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004413-73079.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004413-40780.png)
A.720 | B.120 | C.24 | D.6 |
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004411-31854.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004411-64505.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004411-89582.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004411-64505.png)
A.有最大值2,最小值1 | B.有最大值2,无最小值 |
C.有最大值1,无最小值 | D.无最大值,无最小值 |
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004409-58252.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818004410-60342.png)
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