試題分析:(I)理解
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021467638.png)
且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021482662.png)
的意義,代入后利用函數(shù)的性質(zhì)求解; (Ⅱ)通過(guò)表格得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240220218571301.png)
,再運(yùn)用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021857547.png)
為增函數(shù)建立不等式,導(dǎo)出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021872642.png)
,運(yùn)用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021903775.png)
即可. (Ⅲ)判斷
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021919415.png)
即運(yùn)用反證法證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021935590.png)
,如果
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021950740.png)
使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021966644.png)
則利用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021981665.png)
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021981591.png)
為增函數(shù)一定可以找到一個(gè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021997439.png)
,使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022028833.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022028223.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022044724.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021935590.png)
對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022091688.png)
成立;同樣用反證法證明證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022106573.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022091688.png)
上無(wú)解;從而得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022044724.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022200585.png)
對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022091688.png)
成立,即存在常數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022231515.png)
,使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022044724.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022262736.png)
,有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022278700.png)
成立,選取一個(gè)符合條件的函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022309816.png)
判斷
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022309399.png)
的最小值是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021810262.png)
,由上面證明結(jié)果確定
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021810262.png)
即是符合條件的所有函數(shù)的結(jié)果.
試題解析:(I)因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022356666.png" style="vertical-align:middle;" />且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022371683.png)
,
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240220223871072.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022403566.png)
是增函數(shù),所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022418434.png)
2分
而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240220224341045.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022403566.png)
不是增函數(shù),而
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022465513.png)
是增函數(shù)時(shí),有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022481432.png)
,所以當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022465513.png)
不是增函數(shù)時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021794426.png)
.
綜上得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021794426.png)
4分
(Ⅱ) 因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022356666.png" style="vertical-align:middle;" />,且
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240220225591190.png)
,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022574986.png)
,
同理可證
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240220225901011.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022605903.png)
三式相加得
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021872642.png)
6分
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022637596.png" style="vertical-align:middle;" />所以
而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022668494.png)
,所以
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022699813.png)
8分
(Ⅲ) 因?yàn)榧?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022715977.png" style="vertical-align:middle;" /> 且存在常數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022715312.png)
,使得任取
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022044724.png)
,存在常數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022715312.png)
,使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022793592.png)
對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022091688.png)
成立
我們先證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021935590.png)
對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022091688.png)
成立
假設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021950740.png)
使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021966644.png)
,
記
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022902495.png" style="vertical-align:middle;" />是二階增函數(shù),即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021981591.png)
是增函數(shù).
所以當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022933404.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240220229491040.png)
,所以
所以一定可以找到一個(gè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021997439.png)
,使得
這與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022793592.png)
對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022091688.png)
成立矛盾 11分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021935590.png)
對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022091688.png)
成立
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022044724.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021935590.png)
對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022091688.png)
成立
下面我們證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022106573.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022091688.png)
上無(wú)解
假設(shè)存在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022023136449.png)
,使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022023151607.png)
,
則因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022902495.png" style="vertical-align:middle;" />是二階增函數(shù),即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021981591.png)
是增函數(shù)
一定存在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240220231981269.png)
,這與上面證明的結(jié)果矛盾
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022106573.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022091688.png)
上無(wú)解
綜上,我們得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022044724.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022200585.png)
對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022091688.png)
成立
所以存在常數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022231515.png)
,使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022044724.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022262736.png)
,有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022278700.png)
成立
又令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022309816.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022200585.png)
對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022091688.png)
成立,
又有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022023370755.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022403566.png)
上是增函數(shù),所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022023401648.png)
,
而任取常數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022023417413.png)
,總可以找到一個(gè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022023432454.png)
,使得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022933404.png)
時(shí),有
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022022309399.png)
的最小值為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824022021810262.png)
. 14分