設定義域為

的函數(shù)
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(

為實數(shù))。
(1)若

是奇函數(shù),求

的值;
(2)當

是奇函數(shù)時,證明對任何實數(shù)

都有

成立.
(1)

,(2)證明過程詳見解析.
試題分析:本題考查函數(shù)的奇偶性和函數(shù)最值.考查學生的計算能力和綜合分析問題和解決問題的能力.第一問,利用函數(shù)的奇函數(shù)的性質

,列出表達式,化簡整理得出關于

的恒等式,得出

和

的值;第二問,證明恒成立問題,經過分析題意,只需證明

,所以只需求出
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和

,

是通過配方法求出的,

是通過分離常數(shù)法求出的.
試題解析:(1)(法一)因為

是奇函數(shù),所以

,
即

,∴

,∴

,
∵

,∴

,∴

.(6分)
(法二)因為

是奇函數(shù),所以

,即

對任意實數(shù)

成立.化簡整理得

,這是關于

的恒等式,所以

,所以

(舍)或

.
所以

.(6分)
(2)

,因為

,所以

,

,
從而

;
而

對任何實數(shù)

成立,
所以對任何實數(shù)

、

都有

成立.(12分)
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