已知![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab998be853d1ac2e85c71dc19fc1a3d7.png)
(1)若
单调递增,求
的取值范围;
(2)证明:当
时,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab998be853d1ac2e85c71dc19fc1a3d7.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e30c903d8f8a05332af0b19e7e40df3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82f36662cb2e7e434c341d25976bdbd1.png)
更新时间:2022-05-28 15:21:09
|
相似题推荐
解答题-证明题
|
困难
(0.15)
【推荐1】已知函数
.
(1)当
时,讨论
的单调性;
(2)当
时,若方程
有三个不相等的实数根
,且
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1955f850a56fbd729e8ef999209f098.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/200f24e682c93e02a87f3f9d57dc5d40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5809a06357f94fc7a2156c7e7af1ed2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05b8ec9d4206ea66a02de5c4a1e1e911.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1310a7a80d1f8751a3f8cafe7f8c8b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae9e12d9f9b1dbd7a1ad8fffe752f5e7.png)
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【推荐2】超级细菌是一种耐药性细菌,产生超级细菌的主要原因是用于抵抗细菌侵蚀的药物越来越多,但是由于滥用抗生素的现象不断的发生,很多致病菌也对相应的抗生素产生了耐药性,更可怕的是,抗生素药物对它起不到什么作用,病人会因为感染而引起可怕的炎症,高烧,痉挛,昏迷甚至死亡.某药物研究所为筛查某种超级细菌,需要检验血液是否为阳性,现有n(
)份血液样本,每个样本取到的可能性相等,有以下两种检验方式:(1)逐份检验,则需要检验n次;(2)混合检验,将其中k(
且
)份血液样本分别取样混合在一起检验,若检验结果为阴性,则这份的血液全为阴性,因而这k份血液样本只要检验一次就够了;如果检验结果为阳性,为了明确这k份血液究竟哪几份为阳性,就要对这k份血液再逐份检验,此时这k份血液的检验次数总共为
次.假设在接受检验的血液样本中,每份样本的检验结果是阳性还是阴性都是独立的,且每份样本是阳性结果的概率为p(
).现取其中k(
且
)份血液样本,记采用逐份检验方式,样本需要检验的总次数为
,采用混合检验方式,样本需要检验的总次数为
.
(1)运用概率统计的知识,若
,试求P关于k的函数关系式
;
(2)若P与抗生素计量
相关,其中
,
,…,
(
)是不同的正实数,满足
,对任意的
(
),都有
.
(i)证明:
为等比数列;
(ii)当
时,采用混合检验方式可以使得样本需要检验的总次数期望值比逐份检验的总次数期望值更少,求k的最大值.
参考数据:
,
,
,
,
,
,
,
,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e36bff57bcfa86432b340e25e51d42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c972cbd63decec197aec1bdc306de67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b00f4eb7f1bd2ccefbabf0c1dfa8f69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20c11f6c800b8e0410674a0c6d307d26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e36bff57bcfa86432b340e25e51d42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c972cbd63decec197aec1bdc306de67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d388f32e318b0c7f2d9d10a5c6525b15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90f1ce5bbcc57f96d99d2c4f27cc2e42.png)
(1)运用概率统计的知识,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bd314aee9f06722598766b752fa1e73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cb00792538f7ae7cd3303b465fada7a.png)
(2)若P与抗生素计量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.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/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87a60302649eb940748da818199e55da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13e0971156c0d533e2bf52877ee6a74c.png)
(i)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
(ii)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c8f13606268b633ec8685a36743e8d.png)
参考数据:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0591d9f78b4f4f78c5bd6baaa602ae0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/289ad328bffb5f497153dc0e59939257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4086cd22f9543d857140a3a8e0a7fec9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3616e69114889d5d02099b6598a57136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/401bda9057eb0da43fba681130b558f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3507fab2af0d78ae1fbbfa7d38cb146e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ce9df6d6e8d920d2399bd24ad4a8bf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f4f947bec0390abfadff170732a0338.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d54dec8bafe20965e9a2f685207a23e.png)
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名校
解题方法
【推荐3】某疫苗生产单位通过验血的方式检验某种疫苗产生抗体情况,现有
份血液样本(数量足够大),有以下两种检验方式:
方式一:逐份检验,需要检验n次;
方式二:混合检验,将其中k(
且
)份血液样本混合检验,若混合血样无抗体,说明这k份血液样本全无抗体,只需检验1次;若混合血样有抗体,为了明确具体哪份血液样本有抗体,需要对每份血液样本再分别化验一次,检验总次数为
次.
假设每份样本的检验结果相互独立,每份样本有抗体的概率均为
.
(1)现有7份不同的血液样本,其中只有3份血液样本有抗体,采用逐份检验方式,求恰好经过4次检验就能把有抗体的血液样本全部检验出来的概率;
(2)现取其中k(
且
)份血液样本,记采用逐份检验方式,样本需要检验的总次数为
;采用混合检验方式,样本需要检验的总次数为
.
①若
,求P关于k的函数关系式
;
②已知
,以检验总次数的期望为依据,讨论采用何种检验方式更好?
参考数据:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10b328845a4b1881eee38084d5501224.png)
方式一:逐份检验,需要检验n次;
方式二:混合检验,将其中k(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7399fcd570d1de4057f2059759d18cc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c972cbd63decec197aec1bdc306de67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ea65ed00376f57fd7ec917e69c5bbe9.png)
假设每份样本的检验结果相互独立,每份样本有抗体的概率均为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44fed1be8b7e50f18cb90077d9fce8e4.png)
(1)现有7份不同的血液样本,其中只有3份血液样本有抗体,采用逐份检验方式,求恰好经过4次检验就能把有抗体的血液样本全部检验出来的概率;
(2)现取其中k(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7399fcd570d1de4057f2059759d18cc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c972cbd63decec197aec1bdc306de67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d388f32e318b0c7f2d9d10a5c6525b15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90f1ce5bbcc57f96d99d2c4f27cc2e42.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bd314aee9f06722598766b752fa1e73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2816d5333484a85383df0cd62c7225f0.png)
②已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/868250c34ca12242cf633b5b1ac0f91c.png)
参考数据:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e06a1d75906f9791adc33ea4b69affea.png)
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【推荐1】已知函数
.
(1)若
是
上的单调递增函数,求
的取值范围;
(2)当
满足什么条件时,
恒成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/324fe5b0f9d344069f1fb4a58a84ac5f.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5374bfc463f980d9dab1ffda5f59885.png)
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【推荐2】设函数
,
.
(1)若函数
在区间
是单调函数,求
的取值范围;
(2)设
,证明函数
在区间
上存在最小值
,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2e72d04202a2454ec355e223745647b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1949b35560c8950cef4dc7645fb9f7e3.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c76c7bcfb0dd9e2dc2d9717a0fd06c0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e5be457a03a2b10ef46c1be2ff87b54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fd6a50fe17b747aef94342dcef27b27.png)
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名校
解题方法
【推荐1】已知函数
,
.
(1)若
,证明:
;
(2)若
恒成立,求a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/877e302bb552f3b17d4428b7c48629ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10bbdef421c976962a270a2beabbad91.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb5f421939ee855f25927e7570d82c71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e9c599e8d420006448905acec2b8234.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/495b767764be7fd47876dcebb6f51970.png)
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【推荐2】已知函数
.
(1)若
,且
在
上的最小值为
,求m;
(2)若
有两个不同的极值点
,
(
且
),且不等式
恒成立,求实数t的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87bdc887afc868abf32ed24b2b442807.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9f9d75ebbb4ebb290c4fd3444e1c38b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/354e43caea32568c80e66617d1ce3719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3527bc1a07fb02f88fd1b0768c506e12.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.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/26d8dafc71b106f39f4e15442220897b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba93f19e4328de1bad07d33c7d47cb95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5815762eeea3dc278524ed1ed90cced1.png)
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【推荐3】已知函数
.
(1)讨论函数
的单调性;
(2)已知点
,曲线
在点![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25be20e3724274132cb83b16deaeecfc.png)
处的切线
与直线
交于点
,求
(
为坐标原点)的面积最小时
的值,并求出面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80a254c900435d632b0fc5c632556cd1.png)
(1)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2a29ba49963134a7232fa8574105fc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25be20e3724274132cb83b16deaeecfc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c75aa4c65574bc8fdd71c00b9a4fa7b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4f02028a3847c4807c2d3cf0ea7efb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
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解答题-证明题
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名校
【推荐1】已知函数
在
处的切线方程是
.
(1)求
的单调区间;
(2)如果
且
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bbf2f8c5c8eae0bf124bf343f8c965a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5828873f8369183faf71181cda5b61d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0985b973395bcd371cd1e26d3fcd1c36.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42fd7af568e3d9f444beb0ff41426477.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b76cc836c72b7aa6e7a197d5eb0d3cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/147f89995c5aa07ce7f797c308c9c7d2.png)
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【推荐2】已知函数
与
的图象关于直线
对称.
(1)不等式
对任意
恒成立,求实数
的最大值;
(2)设
在
内的实根为
,
,若在区间
上存在
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61c388166862b3ccfcc7ca749ebe5949.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64263fe2ca48e694c87496d61e63fb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
(1)不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98197299eee2e765e117957b68647f58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/966b60302d80d8613675bb3dd5c03164.png)
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