题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201820-51407.png)
(1)求f (x)在[-1, 1]上的解析式;
(2)证明f (x)在(—1, 0)上时减函数;
(3)当λ取何值时, 不等式f (x)>λ在R上有解?
答案
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201820-34944.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201821-47136.png)
解析
试题分析:(1)当x∈(-1, 0)时, - x∈(0, 1).∴由题意可得f(-x)=
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201821-91571.png)
又f(x)是奇函数,∴f(x)=" -" f (-x) =-
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201820-51407.png)
∵f(-0)= -f(0), ∴f(0)=" 0." 3分
又f(x)是最小正周期为2的函数,∴对任意的x有f(x+2)= f(x).
∴f(-1)=" f(-1+2)=" f(1). 另一面f(-1)="-" f (1), ∴- f(1)=" f(1)" . ∴f(1) = f(-1)=0. 5分
∴f(x)在[-1, 1]上的解析式为 f(x)=
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201820-34944.png)
(2)f (x)在(—1, 0)上时的解析式为
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201821-20596.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201821-75342.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201821-75219.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-28354.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-16507.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-45761.png)
(3)不等式f(x)>λ在R上有解的λ的取值范围就是λ小于f(x)在R上的最大值.…12分
由(2)结论可得,当x∈(-1, 0)时,有-
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-37558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201820-51407.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-34485.png)
又f(x)是奇函数,当x∈(0, 1)时,有
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-34485.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201820-51407.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-37558.png)
∴f(x)在[-1, 1]上的值域是(-
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-37558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-34485.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-34485.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-37558.png)
由f(x)的周期是2;故f(x)在R上的值域是(-
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-37558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-34485.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-34485.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201822-37558.png)
∴λ<
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201821-47136.png)
点评:利用奇偶性求函数解析式问题要注意:(1)在哪个区间求解析式,就设在哪个区间里;(2)转化为已知的解析式进行代入;(3)利用
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201823-20570.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201823-40610.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201823-36648.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201823-20570.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201823-20570.png)
核心考点
试题【定义在R上的函数f(x)是最小正周期为2的奇函数, 且当x∈(0, 1)时, f (x)=.(1)求f (x)在[-1, 1]上的解析式; (2)证明f (x】;主要考察你对函数的相关概念等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201701-66351.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201701-35042.png)
(1)试把方盒的容积
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201701-31545.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201701-35042.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201701-35042.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201701-31545.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201644-16599.png)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201644-93537.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201644-72987.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201644-13627.png)
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201645-36408.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201645-35566.png)
(3)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201645-55088.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201630-56071.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201631-64812.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201559-97211.png)
(1)若y=f(x)在[4,+∞)上为增函数,求实数a的取值范围;
(2)当a=
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201600-49633.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201600-56140.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200204/20200204201552-75613.png)
(1)当a=0时,解不等式f(x)≥g(x);
(2)若存在x∈ R,使得f(x)≥g(x)成立,求实数a的取值范围.
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