题目
题型:解答题难度:简单来源:不详
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170924-99081.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170924-24109.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170924-95926.png)
(Ⅰ)当
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-71541.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-69816.png)
(Ⅱ)函数
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-69816.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-72902.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-71659.png)
(Ⅲ)若对任意给定的
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-34972.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-90198.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170926-97914.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170926-35822.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-71659.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-69816.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170926-78061.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170927-66377.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170927-91012.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170927-91012.png)
解析
试题分析:(Ⅰ)函数f (x)的定义域为
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170927-57168.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170928-58628.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170928-71125.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170928-49513.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170928-35298.png)
故
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-69816.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170926-78061.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170927-66377.png)
(Ⅱ)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170928-40731.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-72902.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170929-79095.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-72902.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170929-29601.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170929-81320.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170929-65535.png)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170930-69186.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170930-91788.png)
故在
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170930-91788.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170930-97667.png)
从而要使
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170930-81175.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170930-66639.png)
只要
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170931-53816.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170931-23275.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170927-91012.png)
(Ⅲ)一次函数
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170924-24109.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170931-65721.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170924-24109.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-90198.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170931-67925.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170931-19202.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170932-37263.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170932-37742.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170932-82104.png)
要使
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-69816.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-90198.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170932-98922.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170932-62714.png)
故
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-69816.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170933-72302.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170933-96836.png)
注意到
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170933-88536.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170933-50363.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170933-63214.png)
∴对任意给定的
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170934-79549.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170934-97531.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170926-97914.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170926-35822.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170931-23275.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170934-21842.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170934-50675.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170934-47045.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170935-47141.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170935-81406.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170935-26140.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170935-66743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170936-62726.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170935-26140.png)
所以,当
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170936-81676.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170936-35929.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170935-26140.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170936-54515.png)
又由
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170937-79344.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170937-17105.png)
∴ 综合①②可知,当
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170927-91012.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170937-21154.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170925-90198.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170926-97914.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170926-35822.png)
点评:导数是研究函数性质的有力工具,研究单调性、极值、最值时不要忘记先求函数的定义域,而不等式恒成立问题,一般转化为函数的最值问题解决,分类讨论时要注意分类标准要不重不漏.
核心考点
试题【已知函数,,(为自然对数的底数).(Ⅰ)当时,求函数的单调区间;(Ⅱ)函数在区间上恒为正数,求的最小值;(Ⅲ)若对任意给定的,在上总存在两个不同的,使得成立,求】;主要考察你对函数定义域等知识点的理解。[详细]
举一反三
设函数
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170912-21847.png)
(Ⅰ) 当
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170912-92280.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170912-45606.png)
(Ⅱ)当
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170912-54311.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170913-54244.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170913-49776.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170913-83374.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170910-12129.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170910-82709.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170911-83555.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170908-21932.png)
(I)求
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170908-49634.png)
(II)若
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170908-49634.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170908-38433.png)
(III)是否存在
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170908-38433.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170908-49634.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170908-38433.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170906-36632.png)
A.[1,+∞) | B.(![]() | C.[![]() | D.(![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20190817/20190817170902-45407.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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