解题方法
1 . 已知函数
.
(1)若
,
,设函数
,请求出
的值域并求证:
;
(2)若
,
,
,记
,且
是一个三角形的三条边长,请写出方程
的所有正整数解的集合;
(3)若
是一个等腰钝角三角形的三条边长且
为最长边,求证:
在
时恒成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b5905520c2d7ba5536552341573fa37.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca27cc54ca0332245f5167488daa3408.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4580cc037c0c760c728cdbb74a8154c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/954e74ff18fc27295263b862e7b559fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6a917e05cfca420bd81408cc7a02133.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e399dbac2fed2f3f99ef9cfce9b5123a.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75d508536d0c182db3e7f81a919793de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6996c86f28de1714e1ccd1c4f77aaa51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b93521270f25a0bcf1618b39808369f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb6f261914d5f3fdf29325d812af540.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a71baf6217604517fd98fa97d0f55b43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4c77befb23ddbca57b9c341f5b9412e.png)
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解题方法
2 . 如果
,记
为区间
内的所有整数.例如,如果
,则
;如果
,则
或3;如果
,则
不存在.已知
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26039f31af6cacb74636a90ae4df9b40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4562f3225c98cf5cb11b47d98c9cc9c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e8a094f3cc5dfbcdf2579830faccab2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c38da3ff65e3aa467094a04c37979d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/377b5d6a1db57c02d6fa64e9e55fe8bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/583b10b1050c1de417cf05733d9943f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30b174dbab52ccacf8fd89bb156ef025.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d88bc255deebebc07d5312cf8c46df4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f2caedee92c1cb1cadcdb1c7bc0261f.png)
A.36 | B.35 | C.34 | D.33 |
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名校
3 . 设全集为
,定义域为
的函数
是关于x的函数“函数组”,当n取
中不同的数值时可以得到不同的函数.例如:定义域为
的函数
,当
时,有
若存在非空集合
满足当且仅当
时,函数
在
上存在零点,则称
是
上的“跳跃函数”.
(1)设
,若函数
是
上的“跳跃函数”,求集合
;
(2)设
,若不存在集合
使
为
上的“跳跃函数”,求所有满足条件的集合
的并集;
(3)设
,
为
上的“跳跃函数”,
.已知
,且对任意正整数n,均有
.
(i)证明:
;
(ii)求实数
的最大值,使得对于任意
,均有
的零点
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b52b4f24969673c863b5aff4fb6751ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2425552313d50a253bfb3cb4e9974ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b52b4f24969673c863b5aff4fb6751ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5182856db60fa5cfda34c97b5748197a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02670179163cffe5070d209066b7aa12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d18ae300954e363c2637120f4f3ef82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45020afb5156159ad42add5537797ee7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b05f8bcb38a5c47e2e8fe9889717fc88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61b4ceef651d43872a078d48092417d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61b4ceef651d43872a078d48092417d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2e77ed55488688257efc354fad8875c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c853fd24a33bd11fbf2d5dba50806d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00108abe0b3ee27e7549f6cc0d86c36f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61b4ceef651d43872a078d48092417d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b52b4f24969673c863b5aff4fb6751ce.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02670179163cffe5070d209066b7aa12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61b4ceef651d43872a078d48092417d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3946552f0f9f048a916879402e4d315a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47efc68941a3be03f5bebbabfbe388fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06e94f0ab8e7418164e0c7481150e6b5.png)
(i)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76d80a81e375bf3c3bdc3603ef7a2a37.png)
(ii)求实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b05f8bcb38a5c47e2e8fe9889717fc88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61b4ceef651d43872a078d48092417d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9fbab6e1a7963d26e1265e1686cba40.png)
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2024高三·全国·专题练习
4 . 下列大小关系正确的是.( )
A.![]() | B.![]() |
C.![]() | D.![]() |
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名校
解题方法
5 . 已知甲植物生长了一天,长度为
,乙植物生长了一天,长度为
.从第二天起,甲每天的生长速度是前一天的
倍,乙每天的生长速度是前一天的
,则甲的长度第一次超过乙的长度的时期是( )(参考数据:取
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6ec1e326713ddcd6dd66a24a809bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294e71b833e7a7a4711b829ff3164fc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4b8503f4706b8321e4e79a87eadea84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/279c21a22d6a7ac8faed319940fe69d5.png)
A.第6天 | B.第7天 | C.第8天 | D.第9天 |
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2024-02-27更新
|
894次组卷
|
7卷引用:山东省济南第一中学等校2024届高三下学期阶段性检测(开学考试)数学试题
名校
解题方法
6 . 临沂一中校本部19、20班数学小组在探究函数的性质时,发现通过函数的单调性、奇偶性和周期性,还无法准确地描述出函数的图象,例如函数
和
,虽然它们都是增函数,但是图像上却有很大的差异. 通过观察图像和阅读数学文献,该小组了解到了函数的凹凸性的概念. 已知定义:设连续函数f(x)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
为凸函数. 对于函数的凹凸性,通过查阅资料,小组成员又了解到了琴生不等式(Jensen不等式):若f(x)是区间
上的凹函数,则对任意的
,有不等式
恒成立(当且仅当
时等号成立). 小组成员通过询问数学竞赛的同学对他们研究的建议,得到了如下评注:在运用琴生不等式求多元最值问题,关键是构造函数.小组成员选择了反比例型函数
和对数函数
,研究函数的凹凸性.
(1)设
,求W=
的最小值.
(2)设
为大于或等于1的实数,证明
(提示:可设
)
(3)若a>1,且当
时,不等式
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12be206d66e65eb92ef08bad8cd8f71d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a7cd59277a15b4d9063be84a40d5541.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4a4ab6155e1fd2c8f9508efa3adcda0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f87a3affc8cd30c21af57157d156c48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c6933733e82337e6d4a95fc2946ff26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2697ef67790838c84cc238a0334c5d47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83aa9d22736190332e01260e5a7803de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29b7a76267b71e6fc828cf2a2e81173d.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21dd60e2cd1a1aae21a9c07820214290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0823f59998a025e80b46881993e89d1.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01262e3dd65728a29f3bbfa584dccede.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7425d1d31f6188375d44137c2b219b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10cda4049695561dab3e0803c3a287fe.png)
(3)若a>1,且当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a89c2336e46cbbe2b978d7d8fcd340be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdc069f6b9d1623e1c06879cef933e42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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2024-02-20更新
|
343次组卷
|
2卷引用:山东省临沂第一中学2023-2024学年高一上学期期末模拟数学试题
7 . 已知函数
,假如存在实数
,使得
对任意的实数
恒成立,称
满足性质
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3992d93b8257ca1c354b4c47d7e7afb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26376c11e57ebc6e98780d8fe466cf9e.png)
A.若![]() ![]() ![]() ![]() |
B.若![]() ![]() ![]() |
C.若![]() ![]() ![]() |
D.若![]() ![]() ![]() ![]() ![]() ![]() |
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解题方法
8 . 已知
,
,
,则下列结论一定成立的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de07daa4c7c7cfcf5241726fa788e6ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/832444aae81de7f780b5e496315cc82e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d625d77f3bfadc66947098bc41233d2.png)
A.若![]() ![]() |
B.若![]() ![]() |
C.若![]() ![]() |
D.若![]() ![]() |
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解题方法
9 . 在数学中,不给出具体解析式,只给出函数满足的特殊条件或特征的函数称为“抽象函数”.我们需要研究抽象函数的定义域、单调性、奇偶性等性质.对于抽象函数
,当
时,
,且满足:
,均有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367936b458618efb6b2eadc843e5d6ba.png)
(1)证明:
在
上单调递增;
(2)若函数
满足上述函数的特征,求实数
的取值范围;
(3)若
,求证:对任意
,都有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c73a98c1b3504e09bfbe0db849b0d24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6571b33b56c6cd88f2f6e091031bcf40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367936b458618efb6b2eadc843e5d6ba.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/639c5f8b7a1a268c904d04356f0d1b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/249a976e88133f3b3733f09137cf5c42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3be9b79f42bbf0de1851607050c3e8d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/219598f1289ddb370d632ea141731d52.png)
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名校
解题方法
10 . 双曲函数是一类与三角函数类似的函数,基本的双曲函数有:双曲正弦函数
,双曲余弦函数
,双曲正切函数
.给出下列四个结论:
①函数
是偶函数,且最小值为2;
②函数
是奇函数,且在
上单调递增;
③函数
在
上单调递增,且值域为
;
④若直线
与函数
和
的图象共有三个交点,这三个交点的横坐标分别为
,
,
,则
.
其中所有正确结论的序号是________________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f45428262907786c0f71f8233820ce2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f64d8edcac9fb4dc0cdc8c952596b722.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f2bdd2bb1b5116dbd9642e734ee3040.png)
①函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3fb37c1744eb0ccb23705a24320d2e0.png)
②函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7901d406f0b87b937ae820be17bc90ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2628e2dd7a988cc80530e739c22b2280.png)
③函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1895e23c9bf91b7b185b4d93e1a16ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2628e2dd7a988cc80530e739c22b2280.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3b1af5d4b930f8989cf63d44768621e.png)
④若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4ff39dd1dfc9caf911ad0d11ba21d66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3fb37c1744eb0ccb23705a24320d2e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7901d406f0b87b937ae820be17bc90ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/291c25fc6a69d6d0ccfb8d839b9b4462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bef6228a20e5bb30673a1ab701c4c3b2.png)
其中所有正确结论的序号是
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2024-01-19更新
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466次组卷
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3卷引用:北京市丰台区2023-2024学年高一上学期期末练习数学试卷