解题方法
1 . 已知函数
.
(1)用单调性定义证明:
在
上单调递增;
(2)若函数
有3个零点
,满足
,且
.
①求证:
;
②求
的值(
表示不超过
的最大整数).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4247d7790d83be16bc74aa5e5d12dd63.png)
(1)用单调性定义证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed6d804ef44bfc64f824b0ccef71765e.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21f8994d83bf4a688c0ab897a5a40fa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a1cc5cfec94bc5686b41b043acdc8ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1310a7a80d1f8751a3f8cafe7f8c8b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d995c5d2e1e0305d805032e18997986a.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f28cbe8f17c4472d8663f9ccbe3b98f6.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59077d1948911b13d68a572eadbca3cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
您最近一年使用:0次
名校
解题方法
2 . 设函数
的定义域为
.若存在常数
,
,使得对于任意
,
成立,则称函数
具有性质
.
(1)判断函数
和
具有性质
?(结论不要求证明)
(2)若函数
具有性质
,且其对应的
,
.已知当
时,
,求函数
在区间
上的最大值;
(3)若函数
具有性质
,且直线
为其图像的一条对称轴,证明:
为周期函数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/809ea2eff71a0de3db640313ad25b7a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/404d068b60dd901194f1684d023212ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58c65d71e57e6e7697e2f627dcd58583.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8073fa685bc10cf01a0128220feac940.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5e661ad31aa4c6d8684923cf904bf1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45d0588454ec8b64bf86578fb90b39e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa7f9b35017daa8b524c5717a355834a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c55351494cd96fed31976fdc5d9c7292.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36d71f015144ffaf1faec94a259b4a06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
您最近一年使用:0次
2021-08-01更新
|
588次组卷
|
3卷引用:北京市第八中学2023~2024学年高一下学期期中练习数学试卷
解题方法
3 . 已知函数
,等差数列
的前
项和为
,记
.
(1)求证:
的图象关于点
中心对称;
(2)若
,
,
是某三角形的三个内角,求
的取值范围;
(3)若
,求证:
.反之是否成立?并请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47665ff46fcf594d4151c3a89707257f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7830b11dc2634eb661673a04287ddc6.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ebbb2eab12b76127cc87304c212cdd6.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/155c5f3a5c55e0c95191c5a893f63062.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca95ba448d33b5e82aa1a3591dc0adf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4615a158e82ab5ff2c3a84f13d1ccda.png)
您最近一年使用:0次
4 . 对于函数
及实数m,若存在
,使得
,则称函数
与
具有“m关联”性质.
(1)若
与
具有“m关联”性质,求m的取值范围;
(2)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
.
求证:
与
不具有“4关联”性.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf557bc0501acbf300fd4ae5993b7242.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4870a0f8fee7a8357094ab4309263752.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99e1ce7071be0743ded4a087fd908eb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b923078510697d5f7f9ea392eb76dd9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d96101eb5dce02c0213ad008413f3066.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
①在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/263b718b5b3cbc27f3e0ef94f4157f5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ede97915bccd6a7b22d7400c30f8adea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
②对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb63478132d4c1fef3c17e591919da83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18db64040b2fa9d65075b41ada928fa6.png)
求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa9c839f85fe048ed0882889e22f5166.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61d2c5422d4b9f8c11a5ad1b62c6bb87.png)
您最近一年使用:0次
2024-01-24更新
|
1190次组卷
|
4卷引用:广东省华南师范大学附属中学2023-2024学年高一上学期期末数学试题
广东省华南师范大学附属中学2023-2024学年高一上学期期末数学试题黑龙江省哈尔滨市第一二二中学校2024届高三下学期校二模考试数学试题河南省郑州市宇华实验学校2024届高三下学期第三次模拟考试数学试题(已下线)压轴题01集合新定义、函数与导数13题型汇总-2
名校
解题方法
5 . “拐点”又称“反曲点”,是曲线上弯曲方向发生改变的点.设
为函数
的导数,若
为
的极值点,则
为曲线
的拐点.
已知曲线C:
.
(1)求C的拐点坐标;
(2)证明:C关于其拐点对称;
(3)设
为C在其拐点处的切线,证明:所有平行于
的直线都与C有且仅有一个公共点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7013f94a41df38d395aaa830559ae31a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2848ae11c9a59b86a60f206f69efcb19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7013f94a41df38d395aaa830559ae31a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7688fe159aac6cd2422b0f834e2b2338.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c087cba6516cfb66c9d346df7e8a24b.png)
已知曲线C:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/497260109e600d68e2a84b20d791de06.png)
(1)求C的拐点坐标;
(2)证明:C关于其拐点对称;
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
您最近一年使用:0次
6 . 已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/143b917df0520097be222accbddf9394.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62559d143b4a977be9990eebcbec539e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)证明:曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79699156efecc21a555e63da6456031a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a551a88ac426439803f564a3bbee04a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
您最近一年使用:0次
7日内更新
|
7709次组卷
|
7卷引用:2024年新课标全国Ⅰ卷数学真题
2024年新课标全国Ⅰ卷数学真题(已下线)2024年高考数学真题完全解读(新高考Ⅰ卷)专题03导数及其应用(已下线)2024年新课标全国Ⅰ卷数学真题变式题16-19湖北省黄冈市浠水县第一中学2023-2024学年高二下学期期末质量检测数学试题(已下线)五年新高考专题09导数及其应用(已下线)三年新高考专题09导数及其应用
名校
7 . 对于三次函数
.定义:①
的导数为
,
的导数为
,若方程
有实数解
,则称点
为函数
的“拐点”;②设
为常数,若定义在
上的函数
对于定义域内的一切实数
,都有
恒成立,则函数
的图象关于点
对称.
(1)已知
,求函数
的“拐点”
的坐标;
(2)检验(1)中的函数
的图象是否关于“拐点”
对称;
(3)对于任意的三次函数
写出一个有关“拐点”的结论(不必证明).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/012429b7101ba0f84e7b45598ed12db9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10acd6d864583617dd3e71240bf0c857.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df1fa6ca9eb7cea9131dad36db6a0ac6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43db00e106c7d08a76a7ba71ca5e63d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5abde3621d19a60d8b0cad42c06d8891.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43db00e106c7d08a76a7ba71ca5e63d1.png)
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03d5f97843013ef486e10f66543562a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)检验(1)中的函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(3)对于任意的三次函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/012429b7101ba0f84e7b45598ed12db9.png)
您最近一年使用:0次
解题方法
8 . 已知函数
.
(1)若
,根据函数单调性的定义证明
在
上单调递减;
(2)由奇函数的图象关于原点对称可以推广得到:函数
的图象关于点
中心对称的充要条件是
.
据此证明:当
时,函数
的图象关于点
中心对称.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f016b1550b070f314cb4e0f6cc36ea3d.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b02f266bd253738e315e84231235f0d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870ebc2f7aabb028024894568d749934.png)
(2)由奇函数的图象关于原点对称可以推广得到:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/827539d066d1b78e7ef8bc1569864971.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efba78e88fe6b52f0d602b3749c6fc49.png)
据此证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bef6d3dd89c1f9696320616f569d1d63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dc1ceb62351e17b3571798d9e3179ec.png)
您最近一年使用:0次
解题方法
9 . 如图,已知直线
,
是
,
之间的一定点并且点
到
,
的距离分别为
,
,
是直线
上一动点,作
,且使
与直线
交于点
.设
.
面积
关于角
的函数解析式
;
(2)画出上述函数的图象;并根据图象求
的最小值;
(3)证明函数
的图象关于
对称.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cdb9d8425d73a68731f30e0c0e22260.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80b53eab97158937f92039c1e133b0f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f285174fbf90a9742de57c1e53224cff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/822ba132ca9dd0d4a050659aef3c9b26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c499d3ef329f85e59fd72dec6f453bbc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fd7229922fbb3ce09dada883f74fbb1.png)
(2)画出上述函数的图象;并根据图象求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fd7229922fbb3ce09dada883f74fbb1.png)
(3)证明函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fd7229922fbb3ce09dada883f74fbb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4120436d6ff0c58109473edc068257c3.png)
您最近一年使用:0次
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解题方法
10 . 已知函数
,
.
(1)当
时,求函数
的定义域;
(2)当
时,判断函数
的奇偶性并证明;
(3)给定实数
且
,试判断是否存在直线
,使得函数
的图象关于直线
对称?若存在,求出
的值(用
表示);若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97de5b263df88cb2439173792f6da4db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be46d7efcc8185eceefd04c33f417478.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b108ab31cc093f03cf48ad65429889e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
(3)给定实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c400a615a16a1662de98dfb4e49d58d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11abb76da45ffa52b47c3a6b9a03ac7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11abb76da45ffa52b47c3a6b9a03ac7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
2024-01-20更新
|
122次组卷
|
3卷引用:广东省广州市番禺区2023-2024学年高二上学期期末教学质量监测数学试题