已知函數(shù)
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(1) 若函數(shù)
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是單調(diào)遞增函數(shù),求實(shí)數(shù)
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的取值范圍;
(2)當(dāng)
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時(shí),兩曲線
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有公共點(diǎn)P,設(shè)曲線
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在P處的切線分別為

,若切線
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與
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軸圍成一個(gè)等腰三角形,求P點(diǎn)坐標(biāo)和
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的值;
(3)當(dāng)
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時(shí),討論關(guān)于
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的方程

的根的個(gè)數(shù)

(1)

(2)
(3)


即

時(shí),函數(shù)

有兩個(gè)零點(diǎn)即方程

有兩個(gè)根;


即
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時(shí),函數(shù)
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有一個(gè)零點(diǎn)即方程

有一個(gè)根;


即
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時(shí),函數(shù)
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沒有零點(diǎn)即方程
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沒有根

(1)

依題,
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在

上恒成立,
法1:

,又

(當(dāng)且僅當(dāng)

,即

時(shí)取等)∴

.
法2:
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,令

,則

在

上恒成立,
由二次函數(shù)

圖象得,
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;


,
綜合
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、
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得
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.…………………………………………………………4分
(2)
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時(shí),

,設(shè)

,

的傾斜角分別為
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,則
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,由于
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,則
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均為銳角,依題,有以下兩種情況:
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
時(shí),

,
此時(shí),

;
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
時(shí),

,
此時(shí),

.……………………………………………………9分
(3)

時(shí),令
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
,

時(shí),

;

時(shí),

∴

在

上遞增,在

上遞減,∴

,
又

時(shí),

;

時(shí),



即

時(shí),函數(shù)

有兩個(gè)零點(diǎn)即方程

有兩個(gè)根;


即

時(shí),函數(shù)

有一個(gè)零點(diǎn)即方程

有一個(gè)根;


即

時(shí),函數(shù)

沒有零點(diǎn)即方程

沒有根

…………………………………………………………14分
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相關(guān)習(xí)題
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知函數(shù)

.
(1)求

的導(dǎo)數(shù)

;
(2)求證:不等式

上恒成立;
(3)求

的最大值。
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知在函數(shù)

的圖象上以
N(1,
n)為切點(diǎn)的切線的傾斜角為

(Ⅰ)求
m、n的值;
(Ⅱ)是否存在最小的正整數(shù)
k,使得不等式

恒成立?如果存在,請(qǐng)求出最小的正整數(shù)
k;如果不存在,請(qǐng)說明理由;
(Ⅲ)(文科不做)求證:
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
(本小題滿分12分)已知函數(shù)

,且函數(shù)

的圖象關(guān)于原點(diǎn)對(duì)稱,其圖象在

處的切線方程為
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(1)求
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的解析式; (2)是否存在區(qū)間
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使得函數(shù)
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的定義域和值域均為
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,且其解析式為f(x)的解析式?若存在,求出這樣的一個(gè)區(qū)間[m,n];若不存在,則說明理由.
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知函數(shù)
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.
(1)當(dāng)
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時(shí),求
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的單調(diào)區(qū)間和極值;
(2)若對(duì)任意
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,
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恒成立,求
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的取值范圍.
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科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
在R上可導(dǎo)函數(shù)
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當(dāng)
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時(shí)取得極大值。當(dāng)
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時(shí)取得極小值,則
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的取值范圍是( )
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知函數(shù)
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的圖像過點(diǎn)P(-1,2),且在點(diǎn)P處的切線恰好與直線
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垂直。
(1)求函數(shù)
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的解析式;
(2)若函數(shù)
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在區(qū)間
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上單調(diào)遞增,求實(shí)數(shù)m的取值范圍。
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
設(shè)函數(shù)
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為奇函數(shù),其圖象在點(diǎn)
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處的切線與直線
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垂直,且在x=-1處取得極值.
(Ⅰ)求
a,
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,
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的值;
(Ⅱ)求函數(shù)
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在
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上的最大值和最小值。
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