名校
解题方法
1 . 已知
是定义在
上的奇函数,且
.
(1)求
和
的值;
(2)判断
在
上的单调性,并证明你的结论;
(3)求证:
的值域为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b01ae79b7370ecef836c2be4b228db53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc30165c18de623d0a3efb961e606d1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be0ec616dbf64b0e1bfce4c84afad5b3.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
(2)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc30165c18de623d0a3efb961e606d1c.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
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2 . 设函数
的定义域为
,对于区间
(
,
),若满足以下两条性质之一,则称
为
的一个“美好区间”.性质①:对任意
,有
;性质②:对任意
,有
.
(1)判断并证明区间
是否为函数
的“美好区间”;
(2)若
(
)是函数
的“美好区间”,试求实数
的取值范围;
(3)已知定义在
上,且图像连续不断的函数
满足:对任意
(
),有
.求证:
存在“美好区间”,且存在
,使得
不属于
的任意一个“美好区间”.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2f15034a908e359bed8b5e0cc467b9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6a46e678bf9d2df5ad4c782b3dc22f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e31b14d5b4da0298a7dea660b03d1066.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bb6324279df94decba955e04ccfa9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c1e560364dea022693928309250f158.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bb6324279df94decba955e04ccfa9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/494d4b56c165f3bd6d41ea80dddc6b71.png)
(1)判断并证明区间
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6c1756b564bf1d998d8179637011c88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb68ccf2d913a83e68df3524263aa8dd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbae0d22d931ac42b565c7990764a2c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b140e221ddf537b8964fff8557cca0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd8ed92f58d44ee590c425bc741195c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)已知定义在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73254f32b6da29ecc32df2e9f87a4c97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6a46e678bf9d2df5ad4c782b3dc22f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/792349a458f6b6d3905775978ee05818.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/070054c0b4182ab7399ed56925844e93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
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3 . ![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b57ed792fb63c756aa4372e501f73cf.png)
(1)证明:
存在唯一的零点
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/270ce4af9a3c5f7987ddef4988ae0a57.png)
(2)若
的零点记为
,设
,求证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b57ed792fb63c756aa4372e501f73cf.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/270ce4af9a3c5f7987ddef4988ae0a57.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70e5ad7a134838f6ee246e606a625f7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eb3c14b2ab08a915682646f3377b7b4.png)
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2023-10-01更新
|
159次组卷
|
3卷引用:福建省漳州实验高级中学2022-2023学年高一创新班上学期期中考试数学试题
福建省漳州实验高级中学2022-2023学年高一创新班上学期期中考试数学试题福建省厦门市厦门二中2023-2024学年高一上学期12月月考数学试题(已下线)专题04 指数函数与对数函数2-2024年高一数学寒假作业单元合订本
4 . 已知函数
.
(1)求函数
的单调减区间;
(2)设
,求证:函数
在
上有唯一零点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7029bd8089800bab0111238b4ed8b38.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcfac854561d24d839fe961dd89ebd65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02e1c9c97de9198d47306216e9961b80.png)
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5 . 已知
且
,函数
.
(1)记
为数列
的前
项和.当
时,试比较
与2024的大小,并说明理由;
(2)当
时,证明:
;
(3)当
且
时,试讨论
的零点个数.
![](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/ff69be9f14b645d71fe4547677db36de.png)
(1)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e65101ee1a5a3540d9359676ba6319a.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/42c98250092857464fbe6cc0707b89ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7416befa4d79b1101a79adb8983c95a.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/950581caec90a28b5fa8f1e81bf21d19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f7a75bcd70f6b1a6d02dbb92e964e1b.png)
(3)当
![](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/09f86f37ec8e15846bd731ab4fcdbacd.png)
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6 . 已知函数
.
(1)当
时,求
的值域;
(2)当
时,设
,求证:函数
有且只有一个零点;
(3)当
时,若实数
使得
对任意实数
恒成立,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3dd9454d93ceba0aabe7fd49940bfe05.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd7126d6d76248996a222631cc9ea93c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/929d7dcd904be9aac64dfc5c68c3539e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7655d9321940385897c723a4f2136c72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaa5e9b6589b0c44b61f17028394b444.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b511bcbe94aa484c0a067891fbf7968.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90de59980f26e4456ff705ca6842400b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da8690b9a30328d99587ef690df5e704.png)
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7 . 已知函数
(
).
(1)若
在
上的最小值为
,求a的值;
(2)证明:
存在唯一零点
且满足
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d00bb85988077e4cc023328a5e0ba9b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d33da711e50e96568facb18cef27165.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7242b2ab643f9470da77e29d043b893.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5c96f02f851f30177f6172b5f592149.png)
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8 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cd8331d5ac755a3e6a7199f7009b87b.png)
(1)求方程
在
上的解集
(2)设函数
,
.
①证明:
在区间
上有且只有一个零点;
②记函数
的零点为
,证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cd8331d5ac755a3e6a7199f7009b87b.png)
(1)求方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc4dc99c6b418baf1c3fe26dc43ed9f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bccd6a6e85bdf500218a3e75b31f3c.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8ed89ab8263c8b8395936f3f062c432.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa004bb9f1f0272f436081ebf431c283.png)
①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f5a90aeba435af22d6bcdb7b91650b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e68d62482d548bcd517188178fd36bc3.png)
②记函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f5a90aeba435af22d6bcdb7b91650b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26c9cec8a8c34da83e265ab7ce8b1281.png)
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2024-03-27更新
|
352次组卷
|
2卷引用:辽宁省大连市第二十四中学2023-2024学年高一下学期5月期中数学试题
名校
解题方法
9 . 已知函数
为奇函数.
(1)求a的值;
(2)设函数
,
i.证明:
有且只有一个零点;
ii.记函数
的零点为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9b073296a8d31514d8b394331df70c2.png)
(1)求a的值;
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b6625dd55e79c5115e6e7299cb83600.png)
i.证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb2faa63899873813748f6a28b8a92e9.png)
ii.记函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb2faa63899873813748f6a28b8a92e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be0b46f954f3590a00fe5a074e8e931b.png)
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2024-02-23更新
|
586次组卷
|
3卷引用:浙江省宁波市鄞州中学2023-2024学年高二下学期期中考试数学试题
(已下线)浙江省宁波市鄞州中学2023-2024学年高二下学期期中考试数学试题浙江省温州市浙南名校联盟2023-2024学年高一下学期寒假返校联考数学试题广东省广州市铁一中学2023-2024学年高一下学期3月月考数学试题
名校
解题方法
10 . “切线放缩”是处理不等式问题的一种技巧. 如:
在点
处的切线为
,如图所示,易知除切点
外,
图象上其余所有的点均在
的上方,故有
. 该结论可通过构造函数
并求其最小值来证明. 显然,我们选择的切点不同,所得的不等式也不同. 请根据以上材料,判断下列命题中正确命题的个数是( )
;
②
;
③
;
④
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3fe2ef17248ee89e1ca43c0db267a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/807d5f1676dc00e9b0af4656ce047170.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ab466aedd6e176088d8dee7bc3e3aaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/807d5f1676dc00e9b0af4656ce047170.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3fe2ef17248ee89e1ca43c0db267a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ab466aedd6e176088d8dee7bc3e3aaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92b3a5e0854a552973617a73ca89a6c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10a4c61536e3e24b760066c88d5762a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff62be512f2e053659ed6e355adc3cc0.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e121b3db6729caa8fade2d606c5abd69.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c72f9fe9af333736b87aaeb5e331d5e5.png)
④
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a0895395eb64cb1d82cb01eedc75820.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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