1 . 已知函数
,
.
(1)当
时,求函数
的极值;
(2)已知实数
.
①求证:函数
有且仅有一个零点;
②设该零点为
,若
图象上有且只有一对点
,
关于点
成中心对称,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42fe84ecdcafb66c2e3a4dd702503729.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/068eeacfc105bcd3cb3e547a9cc3f39c.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c38ada7012b4fd07e9d345c87f346157.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(2)已知实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed0c40f94da9ef7dc21e296235b2326a.png)
①求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
②设该零点为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ad9a4881ebe1a4a566d0fab96d71baa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9938230f82e91cf09f8157b532baaba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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名校
2 . 定义:设
和
均为定义在
上的函数,其导函数分别为
,
,若不等式
对任意
恒成立,则称
和
为区间
上的“友好函数”.
(1)若
和
是“友好函数”,求
的取值范围;
(2)给出两组函数:①
,
;②
,
,分别判断这两组函数是否为
上的“友好函数”.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22add663bd26e87d972a10dc5fd9ada1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2b2c3252d8371afaddf69bcc633f8be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bb6324279df94decba955e04ccfa9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32e54f38c13f26cd12acfbebecc83c8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a10e5c7ac3c66e7e269a36119899ef41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)给出两组函数:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57dff434ff946b8ae010de8d8cfaccb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ec4fa3b5534a782bbddc28c41826d10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77b63e6bf2740efd054cfa0d8c7edc35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b780cf4fb1016e183c8135e31b5953f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870ebc2f7aabb028024894568d749934.png)
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3 . 已知函数
,
.
(1)若
,讨论函数
的单调性;
(2)若
,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd58e16598e6bdb3c35194af69951a2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895938bc4691b6ad48f8b001dfcad102.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074408cfb3eedc559116996d57d5a087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4175c57c61b71897b10583ad32e5e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47c95440ace01be940f1591eed18ab5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63f78ae07b1452e4f9dd8ba93db61d17.png)
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4 . 已知
,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1bb271ea2668cdcb9251d95044f217f.png)
A.![]() |
B.![]() |
C.若方程![]() ![]() |
D.曲线![]() ![]() |
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解题方法
5 . 已知
,
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5eff234212f9564f9e5280817adf5048.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/156a0cac2c3f5dbda1d6ce3320792713.png)
A.函数![]() ![]() | B.![]() ![]() |
C.函数![]() ![]() | D.函数![]() |
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6卷引用:江西省南昌市第十九中学2024届高三下学期第四次模拟考试数学试卷
江西省南昌市第十九中学2024届高三下学期第四次模拟考试数学试卷江苏省南京市六校联合体学校2023-2024学年高二下学期四月联考数学试卷(已下线)模块一 专题4 导数在不等式中的应用A基础卷(高二人教B版)(已下线)模块3 专题1 第1套 小题进阶提升练【高二人教B】重庆市乌江新高考协作体2024届高考模拟监测(一)数学试题河北省承德市2023-2024学年高二年级下学期5月联考数学试题
2024·全国·模拟预测
名校
解题方法
6 . 已知函数
,且
在点
处的切线的斜率为
.设函数
的最大值为
.
(1)求
的值;
(2)求证:
;
(3)若不等式
,求实数
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43e30342cf3719538b05be0ee1deedf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5828873f8369183faf71181cda5b61d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7c629b0c7a5005cd81845ad5c20bd0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a75cc846e65dc041e856dc618583ccea.png)
(3)若不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f435e65eef4aeed564cc6345af190e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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7 . 已知函数
,其中
.
(1)当
时,求
的极值;
(2)讨论当
时函数
的单调性;
(3)若函数
有两个不同的零点
、
,求实数a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/971822ac7125bb76d66139083584263f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10bbdef421c976962a270a2beabbad91.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b108ab31cc093f03cf48ad65429889e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)讨论当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8145758226601870f0366210b150e047.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
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名校
8 . 已知函数
.
(1)讨论
的单调性;
(2)若
恒成立,求
的取值集合;
(3)若存在
,且
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57fb6e0db82c11dbf6d4a022b12dcbd4.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22e38c541dec8fce1d26886e5ef7d21f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb948b7736245b30c44ef9270da0f88f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a78f37a26bad9bcd4165fedc05ce56c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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|
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解题方法
9 . 已知函数
,且
的极值点为
.
(1)求
;
(2)证明:
;
(3)若函数
有两个不同的零点
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db117c5c7215515e036e3a97a01a2b91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36cccdb3b69ec29b9dacfae2f2e70242.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1bd0f17fea5bb6330a209ecba572ed0.png)
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解题方法
10 . 已知函数
有两个零点
.
(1)求实数a的取值范围;
(2)求证:
;
(3)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40fa8f8f5b08ba22c03f57d82b5445f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aca579894dad67bc82cb715fd48e0d70.png)
(1)求实数a的取值范围;
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfcbdd81ba24d15dcb3af31f8942b0ab.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd7b02489f088df9ba0c7eefbd1c6055.png)
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