定義在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155894303.png)
上的函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155909447.png)
對任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155925534.png)
都有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155940852.png)
(
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155956312.png)
為常數(shù)).
(1)判斷
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155956312.png)
為何值時(shí)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155909447.png)
為奇函數(shù),并證明;
(2)設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156081374.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155909447.png)
是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155894303.png)
上的增函數(shù),且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156128529.png)
,若不等式
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156159863.png)
對任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156174433.png)
恒成立,求實(shí)數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156190337.png)
的取值范圍.
(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156206396.png)
,證明過程詳見解析;(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156221436.png)
.
試題分析:本題主要考查抽象函數(shù)奇偶性的判斷和利用函數(shù)單調(diào)性解不等式.考查學(xué)生的分析問題解決問題的能力.考查轉(zhuǎn)化思想和分類討論思想.第一問,用賦值法證明函數(shù)的奇偶性;第二問,利用單調(diào)性解不等式,轉(zhuǎn)化成恒成立問題,再利用二次函數(shù)的性質(zhì)求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156190337.png)
的取值范圍.
試題解析:(Ⅰ)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155909447.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155894303.png)
上為奇函數(shù),則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156284481.png)
, 1分
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156299472.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156330793.png)
,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156206396.png)
. 2分
證明:由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156362836.png)
,令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156377526.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156393713.png)
,
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156284481.png)
,則有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156424642.png)
.即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156440564.png)
對任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156174433.png)
成立,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155909447.png)
是奇函數(shù).
6分
(Ⅱ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240221564861156.png)
7分
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240221565021087.png)
對任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156174433.png)
恒成立.
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155909447.png)
是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022155894303.png)
上的增函數(shù),∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156564726.png)
對任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156174433.png)
恒成立, 9分
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156596723.png)
對任意
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156174433.png)
恒成立,
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156611425.png)
時(shí)顯然成立;
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156627450.png)
時(shí),由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240221566421013.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156767490.png)
.
所以實(shí)數(shù)m的取值范圍是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022156221436.png)
. 13分
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