(本小題滿分14分)
設(shè)函數(shù)

(

).
(1)當(dāng)

時,求

的最小值;
(2)若

,將

的最小值記為

,求

的表達式;
(3)當(dāng)

時,關(guān)于

的方程

有且僅有一個實根,求實數(shù)

的取值范圍.
(1)由已知有:



,……………………………2分
∵

∴

∴當(dāng)

時,

………………………………………………4分
(2)由于

,∴

,
∴ 當(dāng)

時,則

時,

;
當(dāng)

時,則

時,

;
當(dāng)

時,則當(dāng)

時,

;
綜上,

. ……………………………8分
(3)當(dāng)

時,

,
方程

即

,
即方程

在區(qū)間

有且僅有一個實根,……………………………9分
令

,則有:
解法一、①若

∴

………………………………………………………………10分
②

或

綜上所述,當(dāng)

時,方程

在區(qū)間

有且僅有一個實根.……14分
解法二、由零點存在定理得
①

,

,
∴

; …………………………………………………………………10分
②當(dāng)

時,

,此時

,符合題意;
③當(dāng)

時,

,此時

,符合題意;
綜上,當(dāng)

時,方程

在區(qū)間

有且僅有一個實根.……14分
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
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
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
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
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
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