题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205944-28843.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205944-90365.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205944-25546.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205945-88417.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205945-94203.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205945-78916.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205945-61663.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205945-12260.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205944-25546.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205945-88417.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205946-37855.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205946-60240.png)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205946-89202.png)
(2)函数
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205946-25262.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-18092.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-27290.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-10001.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-26945.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205946-25262.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-18092.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205948-21602.png)
解析
试题分析:(1)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205948-69652.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205948-37871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205948-93460.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-27290.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205949-15303.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205949-25918.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205949-16093.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205949-49878.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-10001.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205950-48562.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205950-39222.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-10001.png)
∴当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-27290.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-10001.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-26945.png)
(2)解法一:由(1)可知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205946-25262.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-18092.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205950-14475.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205946-25262.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-18092.png)
设隔离直线的斜率为
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205944-28843.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205950-19590.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205951-96661.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205951-90815.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205951-22526.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205952-25504.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205952-81180.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205952-38072.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205952-13921.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205953-75383.png)
下面证明
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205953-16989.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205953-65584.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205954-77644.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205954-44844.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205954-25825.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205954-66847.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205955-60106.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205949-25918.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205949-16093.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205955-33918.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205955-84137.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205950-48562.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205956-60107.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205955-84137.png)
∴当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-27290.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205955-84137.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-26945.png)
从而
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205956-50863.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205956-95739.png)
∴函数
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205946-25262.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-18092.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205948-21602.png)
解法二: 由(1)可知当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205953-65584.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205957-29760.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-27290.png)
若存在
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205946-25262.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-18092.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205944-28843.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205944-90365.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205957-78734.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205957-89199.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-27290.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205957-35063.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205959-94653.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204210000-95293.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204210000-74942.png)
后面解题步骤同解法一.
点评:求函数极值要首先确定定义域,通过导数等于零找到极值点,但要说明是极大值还是极小值,第二问中将不等式恒成立问题转化为求函数最值问题,这种转化思路是函数综合题中常用的思路,其中找到函数
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205946-25262.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205947-18092.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205950-14475.png)
核心考点
试题【若存在实常数和,使得函数和对其定义域上的任意实数分别满足:和,则称直线为和的“隔离直线”.已知,为自然对数的底数).(1)求的极值;(2)函数和是否存在隔离直线】;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/20200204/20200204205938-26041.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205853-12176.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205854-27820.png)
(1)若函数
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205854-81176.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205854-39835.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205854-58789.png)
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205855-76627.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205854-81176.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205855-75820.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205854-58789.png)
(3)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205855-43606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205855-83324.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205825-77089.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205826-97131.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205826-58945.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205826-14440.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205826-87092.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205827-26070.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205801-84610.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205801-72791.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205801-49122.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205802-44335.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205801-84610.png)
①函数
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205802-64638.png)
②函数
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205802-73735.png)
③若函数
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205802-41201.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205803-30888.png)
④设
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205803-54593.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205803-77429.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205803-33053.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205804-61807.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205804-33699.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205804-89115.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205804-31710.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205805-34166.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205803-54593.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205805-91015.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204205805-82012.png)
以上命题中为真命题的是
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