【题文】给出下列四个命题:①函数有最小值是;②函数的图象关于点对称;③若“且”为假命题,则、为假命题; ④已知定义在上的可导函数满足:对,都有成立,若当时,,则
题型:难度:来源:
【题文】给出下列四个命题:
①函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021739-53990.png)
有最小值是
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021740-79375.png)
;
②函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021741-34629.png)
的图象关于点
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021741-52729.png)
对称;
③若“
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021741-80250.png)
且
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021742-91396.png)
”为假命题,则
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021741-80250.png)
、
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021742-91396.png)
为假命题;
④已知定义在
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021742-62528.png)
上的可导函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021743-89843.png)
满足:对
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021743-87701.png)
,都有
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021744-76896.png)
成立,
若当
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021744-47456.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021745-94917.png)
,则当
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021745-64451.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021745-94917.png)
.
其中正确命题的序号是
.
答案
【答案】①②④.
解析
【解析】
试题分析:对于命题①,
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021746-82924.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021746-29474.png)
,当且仅当
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021747-50295.png)
,即当
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021748-60660.png)
时,上式取等号,即函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021739-53990.png)
有最小值
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021740-79375.png)
,故命题①正确;对于命题②,由于
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021748-68158.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021749-73019.png)
,故函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021741-34629.png)
的图象关于点
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021741-52729.png)
对称,故命题②正确;对于命题③,若“
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021741-80250.png)
且
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021742-91396.png)
”为假命题,则
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021741-80250.png)
、
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021742-91396.png)
中至少有一个是假命题,故命题③错误;对于命题④,由于函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021749-82606.png)
是奇函数,当
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021744-47456.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021745-94917.png)
,即函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021749-82606.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021749-92347.png)
上单调递增,由奇函数的性质知,函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021749-82606.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021750-67524.png)
上也是单调递增的,即当
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021745-64451.png)
时,仍有
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021745-94917.png)
,故命题④正确,综上所述,正确命题的序号是①②④.
考点:1.基本不等式;2.三角函数的对称性;3.复合命题;4.函数的奇偶性与单调性
举一反三
【题文】给出下列四个命题:
①函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021728-39089.png)
有最小值是
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021729-39937.png)
;
②函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021729-67958.png)
的图象关于点
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021729-11498.png)
对称;
③若“
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021730-68965.png)
且
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021730-58307.png)
”为假命题,则
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021730-68965.png)
、
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021730-58307.png)
为假命题;
④已知定义在
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021731-57640.png)
上的可导函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021731-10770.png)
满足:对
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021731-16768.png)
,都有
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021732-45301.png)
成立,
若当
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021732-21588.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021732-44057.png)
,则当
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021733-46401.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021732-44057.png)
.
其中正确命题的序号是
.
【题文】已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021702-28012.png)
定义在R上的奇函数,当
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021703-23924.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021703-65225.png)
,给出下列命题:
①当
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021704-44820.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021704-32453.png)
②函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021702-28012.png)
有2个零点
③
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021704-81643.png)
的解集为
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021705-94069.png)
④
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021705-30623.png)
,都有
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021705-59947.png)
其中正确的命题是
.
【题文】已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021640-76932.png)
,则函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021641-62028.png)
的值域为
.
【题文】已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021611-62616.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021611-57472.png)
上为减函数,则实数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021612-53683.png)
的取值范围是
.
【题文】已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021549-64950.png)
,则函数
![](http://img.shitiku.com.cn/uploads/allimg/20200326/20200326021549-21050.png)
的值域为
.
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