1 . 已知函数
.
(1)讨论
的单调性;
(2)若
,讨论
的零点个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1992673f2428acad25b02245ce76d589.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
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2 . 若函数
存在零点
,函数
存在零点
,使得
,则称
与
互为亲密函数.
(1)判断函数
与
是否为亲密函数,并说明理由;
(2)若
与
互为亲密函数,求
的取值范围.
附:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d35b13df9d8831bb4368e7036488675d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6db18e638db2fb367cfe10bfaee37229.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13e60075f5d53066c03f106346dada26.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fe2f63cdc7606986d6250facf20ad1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbfd7245d512a98d9105f843c094c7e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
附:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c292239a48d1475428eeb9863d5dceb.png)
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3 . 已知函数
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93208bc770714ae8311ab2ba6274ea8d.png)
A.存在![]() ![]() ![]() |
B.对任意![]() ![]() ![]() |
C.对任意![]() ![]() ![]() |
D.存在![]() ![]() ![]() |
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解题方法
4 . 已知函数
在
上恒成立,则实数a的取值范围为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3ad569559e82c28a6f0e8a131b3f442.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48350c9f896c18a64f27867ca81c9be2.png)
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5 . 已知函数
在
上有且仅有一个零点,则实数
的取值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a644d7a17e0c420e8bb5aa0ea1ce4191.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71163f419555f2ed76075c8ff659fbfc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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6 . 已知函数
,
.
(1)当
时,求函数
的在点
处的切线;
(2)若函数
在区间
上单调递减,求
的取值范围;
(3)若函数
的图象上存在两点
,
,且
,使得
,则称
为“拉格朗日中值函数”,并称线段
的中点为函数的一个“拉格朗日平均值点”.试判断函数
是否为“拉格朗日中值函数”,若是,判断函数
的“拉格朗日平均值点”的个数;若不是,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/595e513cd2f3cf78c51ec868fd8b32a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10bbdef421c976962a270a2beabbad91.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5828873f8369183faf71181cda5b61d2.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a89be1009f96de083175f681f6ae1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26d8dafc71b106f39f4e15442220897b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01f8dca2e85a1231ca1a20d5e35739cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
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解题方法
7 . 已知函数
,
,下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67b647867d91de52884cea8c492fb0f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08b29e5cc1e369f351eb1b505919f6ab.png)
A.函数![]() ![]() ![]() |
B.令![]() ![]() |
C.若![]() ![]() |
D.若![]() ![]() ![]() |
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8 . 将函数
的图象向右平移
个单位长度,得到函数
的图象,则函数
的零点个数为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4201223cfcf158b600f5c72407e32b12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037fb348109dc2063a268b10eb925a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2660bdb91f09c1a2a719bb54539f59f1.png)
A.1 | B.2 | C.3 | D.4 |
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2024-06-15更新
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2卷引用:湖南省娄底市第三中学2023-2024学年高二下学期5月月考数学试题
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9 . 帕德近似是利用分式有理函数逼近任意函数的一种方法,定义分式函数
为
的
阶帕德逼近,其分子是m次多项式,分母是n次多项式,且满足
,
,
,…,
时,
为
在
处的帕德逼近.
(1)求函数
在
处的
阶帕德逼近
;
(2)已知函数
.
①讨论
的单调性;
②若
有3个不同零点
,
,
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bbfee67d1c1cb26b67ef0fe3169e6dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dd909ba3e40f03e2f58a4eed2e05f3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adcb8c6a69df1a0deaba265e204d5f99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047a8c1ed551fccee1c1848746c5f282.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72029562177dfc99a171c9013eb90227.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4573475f70860a3d99b92a329d0d07f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9966dfe9109671c587892bd32f0b6699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6b1868d9850b7103e1326eb001dfbce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58de4362237a2a719cde7c9049903da0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9966dfe9109671c587892bd32f0b6699.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2adc63ce466d20e93f7d09d8b0bf9076.png)
①讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.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/d79fc0ce080b8ad8b63ba63259c680b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76e03829b4bbd1148c7f479ca409d436.png)
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解题方法
10 . 函数
.
(1)若
,求函数
的最大值;
(2)若
在
恒成立,求实数m的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c08ff847d1d293df5cf6835129bd00ea.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecc9920abcee41ad09f346eeb981b9d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4633de9335d15d7685bdecb007a3678c.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4666f9da4fd272fcb311a50420b924ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b14d0c440da20a2472608f1eec52eb8.png)
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