已知
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其中
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是自然對(duì)數(shù)的底 .
(1)若
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在
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處取得極值,求

的值;
(2)求

的單調(diào)區(qū)間;
(1)
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;(2)當(dāng)
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時(shí),

的減區(qū)間是

;當(dāng)

時(shí),

的減區(qū)間是
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,增區(qū)間是
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.
試題分析:(1)函數(shù)在

處取得極值即
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可求解

的值;(2)首先考慮函數(shù)的定義域,對(duì)函數(shù)求導(dǎo)得

,再對(duì)實(shí)數(shù)

進(jìn)行分類討論分別求單調(diào)區(qū)間,分類時(shí)要做到不重不漏.
試題解析:(1 )

.
由已知

, 解得

.
經(jīng)檢驗(yàn),

符合題意. 3分
(2)

.
1)當(dāng)

時(shí),

在

上是減函數(shù). 5分
2)當(dāng)

時(shí),

.
①若

,即

,
則

在

上是減函數(shù),在

上是增函數(shù);
②若

,即

,則

在

上是減函數(shù). 10分
綜上所述,當(dāng)

時(shí),

的減區(qū)間是

,
當(dāng)

時(shí),

的減區(qū)間是

,增區(qū)間是

. 12分
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
解不等式
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事實(shí)上:對(duì)于

有

成立,當(dāng)且僅當(dāng)

時(shí)取等號(hào).由此結(jié)論證明:

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
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(1)當(dāng)

時(shí),求曲線

在

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
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
的單調(diào)區(qū)間;
(3)在(2)的條件下,設(shè)函數(shù)

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
[1,2],

[0,1],使

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
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
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
在曲線

(

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
和點(diǎn)

處的切線互相平行,則
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
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
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