設函數(shù)
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,已知
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和
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為
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的極值點.
(Ⅰ)求
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和

的值;
(Ⅱ)討論函數(shù)

的單調(diào)性;
(Ⅲ)設
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,比較
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與
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的大小.
(1)
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,

.
(2)
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在

和

上是單調(diào)遞增的;在

和

上是單調(diào)遞減的.
(3)(1)
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且

時
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(2)
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或
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時,
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(Ⅰ)因為


,
又
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和
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為
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的極值點,所以
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,
因此
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解該方程組得
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,

.
(Ⅱ)因為
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,
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,所以

,
令

,解得
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,
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,
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.
因為當
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
時,
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;
當
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時,
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.
所以
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在
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和
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上是單調(diào)遞增的;在

和

上是單調(diào)遞減的.
(Ⅲ)由(Ⅰ)可知

,
故
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,令
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,則
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.
令
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,得
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,因為
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時,
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,
所以
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在
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上單調(diào)遞減.故
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時,
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;
因為
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時,
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,所以
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在
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上單調(diào)遞增.
故
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時,

.
所以對任意
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,恒有
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,又
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時,
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,
因此
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且
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時
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,
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或
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時
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,
所以, (1)
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且
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時
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(2)
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或
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時,
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【注:】按以下做法不扣分(以下是高考命題人給的原解)這種解法不太嚴謹,但也被大部分人所接受
(Ⅲ)由(Ⅰ)可知

,
故
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,令

,則

.
令

,得
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,因為
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時,

,
所以
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在
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上單調(diào)遞減.故

時,

;
因為

時,

,所以

在

上單調(diào)遞增.
故

時,

.
所以對任意

,恒有

,又

,因此

,
故對任意

,恒有

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科目:高中數(shù)學
來源:不詳
題型:解答題
已知
(1)當
a=1時,試求函數(shù)

的單調(diào)區(qū)間,并證明此時方程

=0只有一個實數(shù)根,并求出此實數(shù)根;
(2)證明:

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科目:高中數(shù)學
來源:不詳
題型:單選題
在R上可導函數(shù)

當

時取得極大值。當

時取得極小值,則

的取值范圍是( )
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科目:高中數(shù)學
來源:不詳
題型:解答題
已知函數(shù)

(I)(i)求函數(shù)
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的圖象的交點A的坐標;
(ii)設函數(shù)

的圖象在交點A處的切線分別為

是否存在這樣的實數(shù)a,使得

?若存在,請求出a的值和相應的點A坐標;若不存在,請說明理由。
(II)記
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上最小值為F(a),求

的最小值。
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科目:高中數(shù)學
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求函數(shù)
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在

處的導數(shù);
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科目:高中數(shù)學
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設函數(shù)
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
為奇函數(shù),其圖象在點

處的切線與直線

垂直,且在x=-1處取得極值.
(Ⅰ)求
a,
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,

的值;
(Ⅱ)求函數(shù)
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在

上的最大值和最小值。
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科目:高中數(shù)學
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題型:解答題
已知函數(shù)

,曲線

在點x=1處的切線
l不過第四象限且斜率為3,又坐標原點到切線
l的距離為
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,若

時,

有極值.
(I) 求a、b、c的值;
(II) 求

在[-3,1]上的最大值和最小值.
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科目:高中數(shù)學
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題型:解答題
已知

,

.
(I)若

,求函數(shù)

在區(qū)間

的最大值與最小值;
(II)若函數(shù)

在區(qū)間

和

上都是增函數(shù),求實數(shù)

的取值范圍.
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科目:高中數(shù)學
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(本小題滿分12分)已知函數(shù)
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.

(1)當
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時,求函數(shù)

的單調(diào)區(qū)間和極值;
(2)當

時,若對任意

,均有

,求實數(shù)

的取值范圍;
(3)若

,對任意
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、
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,且
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,試比較
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與
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的大小.
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