题目内容
本小题满分10分
已知二次函数
(其中
).
(1)若函数
为偶函数,求
的值;
(2)当
为偶函数时,若函数
,指出
在
上单调性情况,并证明之.
已知二次函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954121731.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954136368.png)
(1)若函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954152447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954183283.png)
(2)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954152447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954214675.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954230442.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954261533.png)
(1)
;(2)见解析。
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954277369.png)
本试题主要是考查了二次函数的奇偶性和函数的单调性的运用。
(1)
为偶函数,即对任意
,
即
,即
对任意
恒成立,得到a的值为零。
(2)由(1),若
为偶函数,则
,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232349544801248.png)
当
时,
在
上单调递减,在
上单调递增,然后结合定义法证明。
解:(1)
为偶函数,即对任意
,
……………2分
即
,即
对任意
恒成立 ……………3分
……………4分
(2)由(1),若
为偶函数,则
,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232349544801248.png)
当
时,
在
上单调递减,在
上单调递增,证明如下: ……………5分
设任意
,且![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234955306429.png)
……………7分
,且
,
,即![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234955431549.png)
,即
即![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234955509727.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234955525195.png)
在
上单调递减 ……………9分
同理,可得
在
上单调递增 ……………10分
(1)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954152447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954308433.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954339630.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954355788.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954386506.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954308433.png)
(2)由(1),若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954152447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954448367.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232349544801248.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954495688.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954230442.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954729612.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954760677.png)
解:(1)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954152447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954308433.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954339630.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954355788.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954386506.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954308433.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954916395.png)
(2)由(1),若
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954152447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954448367.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232349544801248.png)
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954495688.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954230442.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954729612.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954760677.png)
设任意
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234955150829.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234955306429.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232349553383120.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234955353864.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234955306429.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234955400679.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234955431549.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232349554471195.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234955478791.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234955509727.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234955525195.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954230442.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954729612.png)
同理,可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954230442.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823234954760677.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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