题目
题型:解答题难度:一般来源:不详
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051328-42924.png)
(Ⅰ)判断f(x)在
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051328-56343.png)
(Ⅱ)若集合A="{y" | y=f(x),
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051328-31979.png)
(Ⅲ)若存在实数a、b(a<b),使得集合{y | y=f(x),a≤x≤b}=[ma,mb],求非零实数m的取值范围.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051328-56343.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051329-24341.png)
解析
(1)由函数单调性的定义出发,给出证明.
(2)由x的范围算出f(x)的值域.再讲两个集合A和B进行比较.
(3)由前面单调性及函数特征的分析可知,0和1作为分类讨论的两个分界点分别讨论.
解:(1)f(x)在
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051328-56343.png)
∵x≥1时,f(x)=1-
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051329-27349.png)
f(x1)- f(x2)=(1-
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051329-27435.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051329-58454.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051329-15504.png)
∵x1x2>0,x1-x2<0 ∴
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051330-38862.png)
∴f(x)在
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051328-56343.png)
(2)证明f(x)在
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051330-94966.png)
(3)∵a<b,ma<mb,∴m>0 ∵f(x)≥0, ∴ma≥0,又a≠0,∴a>0
1° 0<a<b≤1,由图象知,f(x)当x
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051330-61520.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051330-41771.png)
2° 0<a<1<b,这时f(1)=0,则ma=0,而ma>0 这亦与题设不符;
3° 1≤a<b,f(x)当x
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051330-61520.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051330-43103.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051331-30220.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051331-10064.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051331-62522.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051331-26942.png)
综上可知
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051329-24341.png)
核心考点
试题【已知函数(Ⅰ)判断f(x)在上的单调性,并证明你的结论;(Ⅱ)若集合A="{y" | y=f(x),},B=[0,1], 试判断A与B的关系;(Ⅲ)若存在实数a】;主要考察你对函数的单调性与最值等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051326-98925.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051326-51726.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051327-80677.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051322-24521.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051322-50880.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051323-80294.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051323-58436.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051323-98772.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051319-23648.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051319-56247.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051319-91412.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051319-79269.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051320-76564.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051320-76564.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051318-75130.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051315-84708.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051315-88614.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051315-91825.png)
(1)求f (1)、f (-1)的值;
(2)判断f (x)的奇偶性,并说明理由;
(3)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051315-66527.png)
![](http://img.shitiku.com.cn/uploads/allimg/20190818/20190818051316-62087.png)
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