1 . 若函数
在定义域区间
上连续,对任意
恒有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1cfb878da573f79cb25c8df42b7dd8e.png)
,则称函数
是区间
上的上凸函数,若恒有
,则称函数
是区间
上的下凸函数,当且仅当
时等号成立,这个性质称为函数的凹凸性.上述不等式可以推广到取函数定义域中的任意n个点,即若
是上凸函数,则对任意
恒有
,若
是下凸函数,则对任意
恒有
,当且仅当
时等号成立.应用以上知识解决下列问题:
(1)判断函数
(
,
),
,
在定义域上是上凸函数还是下凸函数;(只写出结论,不需证明)
(2)利用(1)中的结论,在
中,求
的最大值;
(3)证明函数
是上凸函数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4776c85b79df196f606d3ebf3697fbc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9d6fe21d6ed78bfc1d2b9cc41a766c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1cfb878da573f79cb25c8df42b7dd8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53e3d9d86ac5a0f90301f8952bdc4c90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4776c85b79df196f606d3ebf3697fbc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a7cd59277a15b4d9063be84a40d5541.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4776c85b79df196f606d3ebf3697fbc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f333263260646c494225db8a7476c00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b36203ece868797d7f1b130ec483ebfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdddc0ae56c39e2cc1293ccca368359d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b36203ece868797d7f1b130ec483ebfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6369550920162ee040faa3f81df2345.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73223617c8855826298d435673787a94.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1bd0587f5d6a3b5db9e4a93e0dbc0ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20849c00c47cbdc43f18d53341b6c4e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/588bbf780d49cf4d29802c2e4126f112.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e10dce73bdc1d522ae7cb34805ed3d8.png)
(2)利用(1)中的结论,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92e19e4be18878ebb959be989905330a.png)
(3)证明函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2cbe0018800880fbad883926a7beb77.png)
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解题方法
2 . 已知
,
,
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e14facc6aef16f3732c12631d858e8b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3dde19dc4d3119bd2cbe4f04bc69f31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb8c2a7421c540b94bff235b251b8b58.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
3 . 已知
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/950eb213a607cd3f0c68a2a356e2d2ed.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
4 . 已知
,
,且
,则下列不等式成立的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/132bc768c61ab195768601a0be02222a.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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解题方法
5 . 记
的内角A,B,C的对边分别为a,b,c,且a,b,c成等比数列,以边
为直径的圆的面积为
,若
的面积不小于
,则
的形状为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a9c6a736e6eac98a676fa3232db5a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd4ece8552b387eb20f633c32f8142f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41322821ce31416fdac8dd6e0aa41c71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
A.等腰非等边三角形 | B.直角三角形 | C.等腰直角三角形 | D.等边三角形 |
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解题方法
6 . 已知
,
,且
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5be97cd1c7111b654d87d8fbb63b6a84.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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7 . 已知
,则下列不等式一定成立的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c4c098615c6bc7e6dcf72e5b5201a.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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2024-02-13更新
|
611次组卷
|
5卷引用:湖南省常德市2024届高三上学期期末检测数学试题
湖南省常德市2024届高三上学期期末检测数学试题(已下线)第六套 九省联考全真模拟(已下线)考点7 基本不等式及其应用 --2024届高考数学考点总动员【练】重庆市缙云教育联盟2024届高三下学期3月月度质量检测数学试题(已下线)第03讲 等式与不等式的性质(五大题型)(讲义)
解题方法
8 . 设
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235a330804350ebb314a2a90b9957641.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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9 . 已知
,则下列不等关系正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d100c22435a23e017cfe6f535379d3c.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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10 . 若不相等的两个正数a,b满足
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88e2086ec2625b8cb0e125285357d283.png)
A.![]() | B.![]() | C.![]() | D.![]() |
您最近一年使用:0次