【题文】已知关于的方程在区间上有实数根,则实数的取值范围是
【题文】已知关于的方程在区间上有实数根,则实数的取值范围是
题型:难度:来源:
【题文】已知关于
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092728-60957.png)
的方程
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092728-97184.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092728-21870.png)
上有实数根,则实数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092729-53418.png)
的取值范围是
答案
【答案】
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092728-21870.png)
解析
【解析】当
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092729-52093.png)
时,方程为
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092729-91086.png)
,解得
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092730-20765.png)
,符合;
当
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092730-26992.png)
时,记
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092730-11392.png)
,其中
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092731-23029.png)
。当
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092731-50604.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092731-68928.png)
,所以题目条件等价于函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092730-11392.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092731-78258.png)
内有零点。
当
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092732-33136.png)
时有函数对称轴
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092732-74000.png)
,若
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092732-89678.png)
,即
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092732-58345.png)
,此时
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092732-23459.png)
的零点为
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092733-73014.png)
,不符合。因为
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092733-41912.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092733-59138.png)
,即
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092733-37684.png)
,所以可知对称轴
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092733-88946.png)
,画图可知此时
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092734-92323.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092731-78258.png)
内无零点。
当
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092734-20781.png)
时有函数对称轴
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092734-23684.png)
,此时
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092733-59138.png)
恒成立。因为
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092734-64813.png)
,所以有
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092735-91566.png)
,解得
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092735-82470.png)
。所以此时
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092735-27231.png)
综上可得,
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092735-83199.png)
举一反三
【题文】已知关于
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092707-30899.png)
的方程
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092707-26806.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092708-12590.png)
上有实数根,则实数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092708-47464.png)
的取值范围是
【题文】函数
f(
x)=
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092645-64691.png)
的单调增区间为( )
A.(-∞,3] | B.[3,+∞) | C.[-1,3] | D.[3,7] |
【题文】函数
f(
x)=
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092634-96283.png)
的单调增区间为( )
A.(-∞,3] | B.[3,+∞) | C.[-1,3] | D.[3,7] |
【题文】如果函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092614-38295.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092614-62522.png)
上是减函数,那么实数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092615-25754.png)
的取值范围是( ).
【题文】如果函数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092608-23189.png)
在区间
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092610-92914.png)
上是减函数,那么实数
![](http://img.shitiku.com.cn/uploads/allimg/20200327/20200327092610-39723.png)
的取值范围是( ).
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