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 . 若平面有不共线的五点A,B,C,D,O,记
,
,
,
,满足
.
,
,则
的最小值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14390e9b6b44472bdc7a131133ab39b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87cd14dfc0024459f9d8e594c95c5106.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e07dcf0b16163e0e0e0c0f248466ee7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/148bb713c21aa6a7b124090d763e3cfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e615309190595163291f1ef907deead.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b053159ca8d5e41edc3bb5ab25c6dc5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b012f67a250177419b93548952986c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54fc3f6eb84df9cda382c357a1606184.png)
您最近一年使用:0次
2024高一下·上海·专题练习
3 . 某同学用“五点法”画函数
,
在某一周期内的图象时,列表并填入的部分数据如下表:
(1)请填写上表的空格处,并写出函数
的解析式;
(2)将函数
的图象向右平移
个单位,再所得图象上各点的横坐标缩小为原来的
,纵坐标不变,得到函数
的图象,求
的单调递增区间;
(3)在(2)的条件下,若
在
上恰有奇数个零点,求实数
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0fcd4082d14e5a26c6fccf782576856.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1acd2d5f345bc859faaf9edc1036a6c7.png)
![]() | ![]() | ![]() | ![]() | ||
![]() | 0 | ![]() | ![]() | ![]() | ![]() |
![]() | 0 | 1 | 0 | ![]() | 0 |
![]() | 0 | ![]() | 0 | ![]() | 0 |
(1)请填写上表的空格处,并写出函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)将函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7ca334d2ae1289b70941e6af9e336a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91609d305f182620aff2f5d85cc7e17f.png)
(3)在(2)的条件下,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecdb2704ea6bc44b5a75fb3c8a100353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a3810fc5221a4e2e7095f945bb4a2e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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名校
4 . 对于实数
,用
表示不超过
的最大整数,例如
.已知
,则下列3个命题中真命题的个数为__________ .
(1)函数
是周期函数;
(2)函数
的图像关于直线
对称;
(3)方程
有2个实数根.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25f161c2a3717f1b6c62d0d7dae0b606.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65d23555720f4938189847b3531bd7fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2297ab933ad7b37e272c0ac367f150d.png)
(1)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
(2)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ec25b105130d71d3d529524671b6218.png)
(3)方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f277209e43e8881f3b0ec4b1bc0df082.png)
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名校
5 . 对于分别定义在
,
上的函数
,
以及实数
,若存在
,
使得
,则称函数
与
具有关系
.
(1)若
,
;
,
,判断
与
是否具有关系
,并说明理由;
(2)若
与
具有关系
,求
的取值范围;
(3)已知
,
为定义在
上的奇函数,且满足:
①在
上,当且仅当
时,
取得最大值1;
②对任意
,有
.
判断
与
是否具有关系
,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6795cae2df43a722e1355e9562d93c09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c296e45b84cf67a98939aa7334e7d478.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4413796ac3d5ca067bf70334101f5440.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e3ac1b540727626af78788a8e5f15de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6fc39a0cea8095683ad4a20a2f96f42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9da4f4bf9fb72cabd7afc5a67f6ecf1c.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12959f65e9db83c446c35d3261a33171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd5cdde751120c6deab563a6f7f8cf05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/976581d4a974fe50f9f29d430c1289f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd5cdde751120c6deab563a6f7f8cf05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/677007fe2d220c74cbe3c9f4e9f8ccec.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3510b22fcf848f36cfaf5ff8964ce049.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17f318506aebfa6403ca8177d8db36d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9da4f4bf9fb72cabd7afc5a67f6ecf1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
①在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57a78355986534b6e50bd7cabc9290a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ede97915bccd6a7b22d7400c30f8adea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
②对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb63478132d4c1fef3c17e591919da83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0d3607cac89a65327549f87f20ba1c7.png)
判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bca470d8c4a0a60e9f600d6aa264ad1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccb41d5badc297be2d9b8931991ab7a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1115793a13fb11c9e126cbb1fb2ac879.png)
您最近一年使用:0次
7日内更新
|
145次组卷
|
2卷引用:北京市中关村中学2023-2024学年高一下学期期中调研数学试题
解题方法
6 . 对于定义在R上的连续函数
,若存在常数t(
),使得
对任意的实数x都成立,则称
是阶数为t的回旋函数.
(1)试判断函数
是否是一个阶数为
的回旋函数,并说明理由;
(2)若
是回旋函数,求实数ω的值;
(3)若回旋函数
(
)在[0,1]上恰有2024个零点,求ω的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/476d332663b8fc357c1a3fc85f9fa5cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4e71ab4caeea9e300aa3886ff2ef8ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(1)试判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5c7fcef9e4a32491be482939d21ceca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/798af57938408f6e1fa1493c05242aa9.png)
(3)若回旋函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d1989dff229887fdd3fdda4a9a05c8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4456675a5dbe545462a22cef9aca8fe.png)
您最近一年使用:0次
7 . 对于平面向量
,定义“
变换”:
,其中
表示
中较大的一个数,
表示
中较小的一个数.若
,则
.记
.
(1)若
,求
及
;
(2)已知
,将
经过
次
变换后,
最小,求
的最小值;
(3)证明:对任意
,经过若干次
变换后,必存在
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4fde5542ad04744c14f912648f3aa0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41bd66e602e9c043218806708e943c2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb00071815c94c090a4095b4964fefb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f7bc9573b3a8758511c63731db18183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96340894e8fb63c00d778b4d654d0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f7bc9573b3a8758511c63731db18183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/701ab98a2bf1135cd989822b0738e11d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/484c1b7bc2fc5677406e20180f667200.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0624499e16b73afec432dd1afd6153d.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b162d1d5bfaa7760678ea3d624beb171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5c19921380da55f5f1a00809a34503.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35234a3829d238ea479fef9cec166468.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8aceb3666a9d49ef40c39eac116ccd5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20cfb8c8707c3960bf1fd46b805e481d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a887552671e6d4df390320ee9a36150.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)证明:对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f389ec068eb1d1aa586b79097d70a7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dcd78ec8777a8e6e5b32222cdb15c05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06296b9023c1dca6f44b8297842bef7c.png)
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名校
解题方法
8 . 已知
是边长为1的正六边形边上相异的三点,则
的取值范围是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4ed86beeec5a7e4201187c8d7681fc6.png)
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9 . 如图,设
是平面内相交成
的两条射线,
分别为
同向的单位向量,定义平面坐标系
为
仿射坐标系,在
仿射坐标系中,若
,则记
.
仿射坐标系中
①若
,求
;
②若
,且
与
的夹角为
,求
;
(2)如上图所示,在
仿射坐标系中,B,C分别在
轴,
轴正半轴上,
分别为BD,BC中点,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46c092d69df76faf1e2133dc96b466ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43a998a7d4d980e848ee050b706480ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad59ee7969f2a082ed53bdf0aaa748ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46c092d69df76faf1e2133dc96b466ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4446f654cf8401b640cc79c2b97af5db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4446f654cf8401b640cc79c2b97af5db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3def4278aef3c2c3aa64386584e5df65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1826aa6f667b181d7aabc06e35365308.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4446f654cf8401b640cc79c2b97af5db.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7b1fc6efbb1fe3d949bf100925cdf34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5455bdb43226a925e13da2df0f233be6.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c5b0bb8bf0236fde97d668f40fd404a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1a63ab608517bb10aa036783dfb51f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d66c03d4ca06819a6ce7fc8ea6de0f0.png)
(2)如上图所示,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/479feca6887a5b30b7142c665cc61e9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03da507737fe5b3211dc2953d6c971c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a462f4899d41997a8ce2df63d0056e4d.png)
您最近一年使用:0次
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解题方法
10 . 若
为锐角三角形,当
取最小值时,记其最小值为
,对应的
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/549f99e6e10e61af2e7734c4d01ea90c.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efa83b393e9337b1d3f399b6cdee2cd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f48c2d9774d952af04ff1b13447ece01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/549f99e6e10e61af2e7734c4d01ea90c.png)
您最近一年使用:0次