题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104015-57676.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104015-57676.png)
(1)若函数f(x)为集合M中的任意一个元素,证明:方程f(x)一x=0只有一个实根;
(2)判断函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-54694.png)
(3)设函数f(x)为集合M中的任意一个元素,对于定义域中任意
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-88033.png)
证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-87534.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-45222.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-68537.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-12743.png)
所以,方程
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-59091.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-23334.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-23334.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-23334.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-94101.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-54824.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-40099.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-23281.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104018-10487.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104018-76971.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-45222.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-68537.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-12743.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104018-90080.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104018-35550.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104018-24164.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104019-27848.png)
解析
试题分析:令
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-45222.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-68537.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-12743.png)
所以,方程
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-59091.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-23334.png)
又由题设①知方程
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-23334.png)
所以,方程
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-23334.png)
(2)易知,
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104019-63639.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104019-40876.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104019-80667.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104019-74527.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104020-93580.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104019-74527.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104020-93580.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104020-35854.png)
即方程
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104020-38837.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104020-79586.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104020-18030.png)
综上可知,
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-94101.png)
(Ⅲ)不妨设
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-54824.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-40099.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104017-23281.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104018-10487.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104018-76971.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-45222.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-68537.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104016-12743.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104018-90080.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104018-35550.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104018-24164.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104019-27848.png)
点评:近几年新课标高考对于函数与导数这一综合问题的命制,一般以有理函数与半超越(指数、对数)函数的组合复合且含有参量的函数为背景载体,解题时要注意对数式对函数定义域的隐蔽,这类问题重点考查函数单调性、导数运算、不等式方程的求解等基本知识,注重数学思想(分类与整合、数与形的结合)方法(分析法、综合法、反证法)的运用.把数学运算的“力量”与数学思维的“技巧”完美结合.
核心考点
试题【本小题满分12分)设M是由满足下列条件的函数f (x)构成的集合:①方程f (x)一x=0有实根;②函数的导数满足0<<1.(1)若函数f(x)为集合M中的任意】;主要考察你对函数的单调性与导数等知识点的理解。[详细]
举一反三
已知函數f(x)=ln+mx2(m∈R)
(I)求函数f(x)的单调区间;
(II)若m=0,A(a,f(a))、B(b,f(b))是函数f(x)图象上不同的两点,且a>b>0,
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104000-68782.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104000-72147.png)
(III)求证
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018104000-77393.png)
已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103943-21435.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103944-48981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103944-63335.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103944-25838.png)
(Ⅰ)解关于
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103944-80214.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103944-12947.png)
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103944-63335.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103944-52852.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103945-72981.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103933-72779.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103934-31377.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103934-14662.png)
(1)用
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103934-31377.png)
(2)若函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103934-10490.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103934-31377.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103920-38437.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103921-18907.png)
(1)若函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103921-32095.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103921-43765.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103921-89342.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103921-50931.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103921-87211.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103922-42920.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103922-99381.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103922-64439.png)
(3)在(Ⅱ)的条件下,若导函数
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103922-64439.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103922-79126.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103922-26022.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103901-41438.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103901-36560.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103902-50713.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103902-34237.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103902-39309.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103902-96424.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103903-46277.png)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103902-96424.png)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103903-66335.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103903-19952.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103903-91840.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191018/20191018103903-62109.png)
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