名校
解题方法
1 . (1)已知
,求证:
;
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6a93645a9c1f5a2961519d74bf51567.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e22bbb6c88cb559f76a89daa17f7084c.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b3e2dc4e933c2bb7a1cb19cee8d0ad7.png)
您最近一年使用:0次
2024高三·全国·专题练习
解题方法
2 . 已知实数
,
满足
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/432d77fe5ad3032d59a237dd94c8a638.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/838b9f49811c77cbf7d12d3af4a63373.png)
您最近一年使用:0次
名校
解题方法
3 . 设连续函数
的定义域为
,如果对于
内任意两数
,都有
,则称
为
上的凹函数;若
,则称
为凸函数.若
是区间
上的凹函数,则对任意的
,有琴生不等式
恒成立(当且仅当
时等号成立).
(1)证明:
在
上为凹函数;
(2)设
,且
,求
的最小值;
(3)设
为大于或等于1的实数,证明:
.(提示:可设
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a7a1783349936cc7254a4a8694c6812.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1bedaf3854b48806b82b3b804451cf8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0fec4d10407498ec4692b33ebe1bb7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1784a3a9dd90c51dab965445d65f08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2697ef67790838c84cc238a0334c5d47.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/008ab9b6200370bd8d534a6317cb88e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7160d93f92089ef36f3dab809d3114b8.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6da13af19b32430759c9c1d1aea894e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b16ad49f62d7362441e3b92efe7f87d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b47ade684a2e49ef6139afe6ab59a29.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1d4c274b53adfbffc4b19e7adbc39d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6694499b581256296277c515f6dcdc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10cda4049695561dab3e0803c3a287fe.png)
您最近一年使用:0次
2024高三·全国·专题练习
解题方法
4 . 已知
为正实数.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59674d6132d140d9a94f24588ad3b181.png)
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名校
5 . 已知函数
.
(1)求函数
的极值;
(2)设函数
有两个极值点
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f86d97d22525157c58a5148cdbf51a2c.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e38e9aa7ea6401f10eae6ef9a6a45c6.png)
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2024-03-03更新
|
348次组卷
|
4卷引用:安徽省六安市2024届高三上学期期末教学质量检测数学试题
安徽省六安市2024届高三上学期期末教学质量检测数学试题(已下线)第五章综合 第二练 数学思想训练(已下线)专题07 函数的极值和最值的应用8种常考题型归类【好题汇编】-备战2023-2024学年高二数学下学期期末真题分类汇编(北师大版2019选择性必修第二册)吉林省长春市实验中学2023-2024学年高二下学期5月期中考试数学试题
6 . 设正整数
,有穷数列
满足
,且
,定义积值![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e36bd11e8ffb70eac461dc4768b840a.png)
(1)若
时,数列
与数列
的S的值分别为
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0a1568c0bf07f285b2e01c3a3a55900.png)
①试比较
与
的大小关系;
②若数列
的S满足
,请写出一个满足条件的![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddcaa1756eafc97696d69068689892c8.png)
(2)若
时,数列
存在
使得
,将
,
分别调整为
,
,其它2个
,令
数列
调整前后的积值分别为
,写出
的大小关系并给出证明;
(3)求
的最大值,并确定S取最大值时
所满足的条件,并进行证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83056c039e255d1ca7e26b756f3a6d98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b99718e1bce4057550e1aef19c82b30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e36bd11e8ffb70eac461dc4768b840a.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b801f41875296c26e893f492af633bea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a422e11339ddc763ada97021f03722a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0a1568c0bf07f285b2e01c3a3a55900.png)
①试比较
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
②若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd5769559be3487868d334c66d130360.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddcaa1756eafc97696d69068689892c8.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fac3649308b528fd56545ba102dc42d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99a58d2f2ef54bddedcb3ca40b1b43bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c973c75e2e9209e2a22e3deb453e0cce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8e57093fadaaa08e9ac73e855221525.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a272adba0f1120109824440f0e252c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/711e95f069a685da11ff70b16504578a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82d03382cb64aca02dd52d8196abb804.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3c5515db76e233bad7f418cfbcbc0b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f021e668a3bb0b84447138c33a6ca188.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33a3fc7a52b6b15e855cd22bdf8d00bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99a58d2f2ef54bddedcb3ca40b1b43bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ab93d9afb14d07b81567d47207c4be0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ab93d9afb14d07b81567d47207c4be0.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19c11e6c8be2cb8384953b3f19f7b77b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
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7 . “让式子丢掉次数”:伯努利不等式
伯努利不等式(Bernoulli’sInequality),又称贝努利不等式,是高等数学的分析不等式中最常见的一种不等式,由瑞士数学家雅各布·伯努利提出:对实数
,在
时,有不等式
成立;在
时,有不等式
成立.
(1)猜想伯努利不等式等号成立的条件;
(2)当
时,对伯努利不等式进行证明;
(3)考虑对多个变量的不等式问题.已知
是大于
的实数(全部同号),证明
伯努利不等式(Bernoulli’sInequality),又称贝努利不等式,是高等数学的分析不等式中最常见的一种不等式,由瑞士数学家雅各布·伯努利提出:对实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6339f512d6f801fde040ae9677056d98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e62a78f2a44f317b65a4d05f0c76a927.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbb83894d7274b0c36842fa7c51cc466.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb986bcbf5c3c17aefc7ac8a1a68b82c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/267a28b9f6d9e9f5b761a94ca2075bb4.png)
(1)猜想伯努利不等式等号成立的条件;
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4360c1b4a506c12bbdce41e73fb74d8.png)
(3)考虑对多个变量的不等式问题.已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/206a6f31229c1b9905aca55c50369c4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c450723559c1574d3a557bfb7e943fd6.png)
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名校
解题方法
8 . 如果函数
满足以下两个条件,我们就称
为
型函数.
①对任意的
,总有
;
② 当
时,总有
成立.
(1)记
,求证:
为
型函数;
(2)设
,记
,若
是
型函数,求
的取值范围;
(3)是否存在
型函数
满足:对于任意的
,都存在
,使得等式
成立?请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
①对任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047056c99b39c70fa40d3c8178e5b631.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/018857ec6e498113b3b12a730d9313da.png)
② 当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfdccde6a17dc78bec232630577f99d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16d5873aa225a83805e1072ef8119b7a.png)
(1)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc540d6c4de05039557cdfe8c78ceeec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19339e3904e9541ff26b30ae5f1242b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e54428f4829c8061f79df9f492305c3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/951b05c96af4f7704de24ac541b3f172.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
(3)是否存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efb679de6747c1a9147225d7b61c436f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbe0c952b97016a6816cfca66e024ef4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acdc6e6a0e6584bea7deb91b0841fa28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fe8e17429b079c4965fae3bef4e6b25.png)
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2024-01-10更新
|
420次组卷
|
2卷引用:上海市静安区2024届高三上学期期末教学质量调研数学试题
名校
9 . 已知等比数列
的各项均为正数,且
.
(1)求数列
的通项公式;
(2)设
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fff2853ed2252ef24654888b3bea2084.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2597e30850f276a5fe5ae97bee20573a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aacabd66c249e6b0b3562056718f7901.png)
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2024-01-10更新
|
1597次组卷
|
3卷引用:辽宁省沈阳市2023-2024学年高三上学期教学质量监测(一)数学试题
名校
解题方法
10 . (1)已知
,
,
,证明:
;
(2)证明:当
,
时,有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/061813f1ec633c5c4c393c4de7938322.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/131ac7eb1e911c9a40e84235bf3742ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/743521dbdd63eb1ccda627d0f695db05.png)
(2)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37c84b49231d0344d0813a7bbd2acdaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a14a2671cb729ab8919ad06008671d69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b49236cb8b41994a11aee4998576dd48.png)
您最近一年使用:0次