【题文】已知函数 且此函数图象过点(1,5).(1)求实数m的值;(2)判断奇偶性;(3)判断函数在上的单调性?并用定义证明你的结论.
题型:难度:来源:
【题文】已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214731-62263.png)
且此函数图象过点(1,5).
(1)求实数m的值;
(2)判断
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214731-29980.png)
奇偶性;
(3)判断函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214731-29980.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214731-64418.png)
上的单调性?并用定义证明你的结论.
答案
【答案】(1)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214732-72678.png)
;(2)见解析;(3)见解析.
解析
【解析】
试题分析:(1)把
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214732-79079.png)
代入函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214732-41908.png)
,可求得
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214732-72678.png)
;
(2)利用奇偶性的定义可得:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214733-94504.png)
,即可得到结论;
(3)函数在
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214733-10898.png)
上单调减,利用单调性的定义证明,取值,作差,变形,定号下结论;
试题解析:(1)把
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214732-79079.png)
代入函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214732-41908.png)
得
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214733-82148.png)
,解得
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214732-72678.png)
(2)由(1)可得:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214734-40184.png)
,所以
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214733-94504.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214732-41908.png)
是奇函数;
(3)函数在
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214733-10898.png)
上单调递减,证明如下:
取
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214734-93488.png)
,则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214734-62134.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214735-87019.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214734-93488.png)
,所以
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214735-65987.png)
,∴
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214736-66995.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214736-46334.png)
,所以
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214736-23135.png)
∴函数在
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214733-10898.png)
上单调递减.
考点:函数性质的综合应用.
举一反三
【题文】设定义在R上的函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214428-51977.png)
,对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214428-20443.png)
有
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214429-15717.png)
,且当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214429-11102.png)
时,恒有
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214430-50724.png)
,
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214430-79344.png)
;
(2)判断该函数的奇偶性;
(3)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214431-34260.png)
时 ,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214428-51977.png)
为单调递增函数.
【题文】设函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214400-96514.png)
.
(Ⅰ)求函数y=f(x)的最小值.
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214400-28009.png)
恒成立,求实数a的取值范围.
【题文】设
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214208-64382.png)
,若函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214209-68658.png)
为单调递增函数,且对任意实数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214209-77312.png)
,都有
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214210-21274.png)
(
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214211-21561.png)
是自然对数的底数),则
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214211-95917.png)
( )
A.1 | B.![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214212-89396.png) | C.3 | D.![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325214212-84131.png) |
【题文】已知命题
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213744-63342.png)
:函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213744-19318.png)
为
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213744-78071.png)
上单调减函数,实数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213745-12120.png)
满足不等式
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213745-66597.png)
.命题
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213745-84545.png)
:当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213746-42487.png)
,函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213746-95160.png)
.若命题
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213744-63342.png)
是命题
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213745-84545.png)
的充分不必要条件,求实数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213747-94029.png)
的取值范围。
【题文】已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213733-57004.png)
的定义域为[
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213733-65800.png)
],部分对应值如下表:
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213733-57004.png)
的导函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213735-65270.png)
的图象如图所示,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213735-38403.png)
下列关于
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213733-57004.png)
的命题:①函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213733-57004.png)
是周期函数;②函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213733-57004.png)
在[0,2]上是减
函数;③如果当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213736-18651.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213733-57004.png)
的最大值是2,那么
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213736-92317.png)
的
最大值是4;④当
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213736-73944.png)
时,函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213737-65801.png)
有4个零点;
⑤函数
![](http://img.shitiku.com.cn/uploads/allimg/20200325/20200325213737-65801.png)
的零点个数可能为0,1,2,3,4。其中正确命题的序号是_____________(写出所有正确命题的序号).
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