已知函數(shù)

與

的圖像都過點

,且它們在點

處有公共切線.
(1)求函數(shù)

和

的表達式及在點

處的公切線方程;
(2)設

,其中

,求

的單調(diào)區(qū)間.
(1)

,

,

;
(2)當

時,F(xiàn)(x)的單調(diào)減區(qū)間是

單調(diào)增區(qū)間是

;
當

時,F(xiàn)(x)沒有單調(diào)減區(qū)間,單調(diào)增區(qū)間是

.
試題分析:(1)因為函數(shù)
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和

有公共的切線,所以切線的斜率相同,又因為它們都過

,所以可以列出方程,求出

;(2)先求導數(shù),求出函數(shù)的定義域,通過討論

的正負,求導求單調(diào)區(qū)間.
試題解析:(1)∵

過點
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∴
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,
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, (2分)
∵

,∴切線的斜率

.
∵

,

(1)
又∵

的圖像過點

∴

(2)
聯(lián)立(1)(2)解得:

(4分)
∴

;切線方程為

,即

∴

,

;切線為:

(6分)
(2)∵

,
∴
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(9分)
①當

時,

, ∵

,∴

又

,∴當

時,

;
當

時,

.
∴

的單調(diào)減區(qū)間是

單調(diào)增區(qū)間是

; (11分)
②當

時,顯然

沒有單調(diào)減區(qū)間,單調(diào)增區(qū)間是

. (13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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
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
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
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
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(Ⅱ)若

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
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
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規(guī)定

其中

,
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
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
(

是正整數(shù),

)的一種推廣.
(Ⅰ) 求

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(Ⅱ)排列數(shù)的兩個性質(zhì):①


,②


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(其中m,n是正整數(shù)).是否都能推廣到
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(
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,
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是正整數(shù))的情形?若能推廣,寫出推廣的形式并給予證明;若不能,則說明理由;
(Ⅲ)已知函數(shù)

,試討論函數(shù)

的零點個數(shù).
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科目:高中數(shù)學
來源:不詳
題型:解答題
己知函數(shù)

.
(I)求f(x)的極小值和極大值;
(II)當曲線y = f(x)的切線

的斜率為負數(shù)時,求

在x軸上截距的取值范圍.
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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)

(1)討論函數(shù)

的單調(diào)性;
(2)若
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時,關(guān)于

的方程
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有唯一解,求
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的值;
(3)當
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時,證明: 對一切

,都有
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成立.
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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)

.
(1)當
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時,求

的單調(diào)區(qū)間;
(2)若函數(shù)
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在
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上無零點,求
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的最小值。
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