【题文】(本题满分14分) 已知函数,其中(Ⅰ)求函数的定义域;(Ⅱ)若对任意恒有,试确定的取值范围.
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【题文】(本题满分14分) 已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100100-99065.png)
,其中
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100101-52179.png)
(Ⅰ)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100101-86321.png)
的定义域;
(Ⅱ)若对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100101-21534.png)
恒有
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100102-35235.png)
,试确定
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100102-45095.png)
的取值范围.
答案
【答案】(Ⅰ)
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100102-79640.png)
;(Ⅱ)
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100103-30814.png)
.
解析
【解析】
试题分析:(Ⅰ)根据对数函数的定义知,满足函数的定义域需满足条件:
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100103-13565.png)
,结合已知条件
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100101-52179.png)
可分两种情况讨论:
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100103-69654.png)
和
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100104-81738.png)
,分别求出其满足的定义域,然后作并集即可;
(Ⅱ)运用变量分离法将问题“对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100101-21534.png)
恒有
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100102-35235.png)
”转化为“
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100104-57674.png)
对
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100104-37966.png)
恒成立”,即
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100105-43992.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100105-31450.png)
,然后结合函数
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100105-22904.png)
的增减性判断其最大值,即可求出
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100102-45095.png)
的取值范围.
试题解析:(Ⅰ) 由
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100103-13565.png)
得,
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100106-90308.png)
,因为
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100106-61280.png)
,所以
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100106-62437.png)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100103-69654.png)
时,定义域为
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100107-32660.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100107-72114.png)
时,定义域为
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100108-15504.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100104-81738.png)
时,定义域为
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100108-39194.png)
;
(Ⅱ)对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100104-37966.png)
恒有
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100108-22374.png)
,即
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100109-16876.png)
对
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100104-37966.png)
恒成立
即
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100104-57674.png)
对
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100104-37966.png)
恒成立
记
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100109-36794.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100104-37966.png)
,则只需
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100109-68993.png)
而
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100109-36794.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100110-93018.png)
上是减函数,所以
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100110-20233.png)
故为
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100103-30814.png)
.
考点:对数函数的定义域;导数在研究函数中的应用.
举一反三
【题文】(本题满分14分) 已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100054-37427.png)
,其中
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100054-12410.png)
(Ⅰ)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100054-33436.png)
的定义域;
(Ⅱ)若对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100054-35346.png)
恒有
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100055-81906.png)
,试确定
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100055-59619.png)
的取值范围.
【题文】(本题满分14分) 已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100037-27143.png)
,其中
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100038-87600.png)
(Ⅰ)求函数
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100038-33310.png)
的定义域;
(Ⅱ)若对任意
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100038-57113.png)
恒有
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100039-70024.png)
,试确定
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100039-94293.png)
的取值范围.
【题文】(满分12分)不用计算器计算:(注:只要有正确的转换,都要给步骤分,不能只看结果)
(1)
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100029-17866.png)
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100029-67180.png)
【题文】(满分12分)不用计算器计算:(注:只要有正确的转换,都要给步骤分,不能只看结果)
(1)
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100024-31596.png)
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100024-73595.png)
【题文】已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100019-41894.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100019-62816.png)
上是减函数,则
![](http://img.shitiku.com.cn/uploads/allimg/20200330/20200330100019-99776.png)
的取值范围是 ( )
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