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專(zhuān)題06函數(shù)與導(dǎo)數(shù)常見(jiàn)經(jīng)典壓軸小題歸類(lèi)目錄01函數(shù)零點(diǎn)問(wèn)題之分段分析法模型 302函數(shù)嵌套問(wèn)題 703函數(shù)整數(shù)解問(wèn)題 1104唯一零點(diǎn)求值問(wèn)題 1405等高線(xiàn)問(wèn)題 1606分段函數(shù)零點(diǎn)問(wèn)題 1907函數(shù)對(duì)稱(chēng)問(wèn)題 2208零點(diǎn)嵌套問(wèn)題 2609函數(shù)零點(diǎn)問(wèn)題之三變量問(wèn)題 3010倍值函數(shù) 3311函數(shù)不動(dòng)點(diǎn)問(wèn)題 3712函數(shù)的旋轉(zhuǎn)問(wèn)題 4013構(gòu)造函數(shù)解不等式 4414導(dǎo)數(shù)中的距離問(wèn)題 4815導(dǎo)數(shù)的同構(gòu)思想 5016不等式恒成立之分離參數(shù)、分離函數(shù)、放縮法 5317三次函數(shù)問(wèn)題 5818切線(xiàn)條數(shù)、公切線(xiàn)、切線(xiàn)重合與垂直問(wèn)題 6219任意存在性問(wèn)題 6720雙參數(shù)最值問(wèn)題 7021切線(xiàn)斜率與割線(xiàn)斜率 7222最大值的最小值問(wèn)題(平口單峰函數(shù)、鉛錘距離) 7523兩邊夾問(wèn)題和零點(diǎn)相同問(wèn)題 7924函數(shù)的伸縮變換問(wèn)題 8125V型函數(shù)和平底函數(shù) 8326曼哈頓距離與折線(xiàn)距離 8701函數(shù)零點(diǎn)問(wèn)題之分段分析法模型1.(2023·黑龍江·高三大慶市東風(fēng)中學(xué)校考期中)設(shè)函數(shù)SKIPIF1<0(其中SKIPIF1<0為自然對(duì)數(shù)的底數(shù)),若函數(shù)SKIPIF1<0至少存在一個(gè)零點(diǎn),則實(shí)數(shù)SKIPIF1<0的取值范圍是A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<02.(2023·湖北·高三校聯(lián)考期中)設(shè)函數(shù)SKIPIF1<0,記SKIPIF1<0,若函數(shù)SKIPIF1<0至少存在一個(gè)零點(diǎn),則實(shí)數(shù)SKIPIF1<0的取值范圍是A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<03.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知函數(shù)SKIPIF1<0(e為自然對(duì)數(shù)的底數(shù))有兩個(gè)不同零點(diǎn),則實(shí)數(shù)SKIPIF1<0的取值范圍是___________.4.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知函數(shù)SKIPIF1<0存在4個(gè)零點(diǎn),則實(shí)數(shù)SKIPIF1<0的取值范圍是__________.02函數(shù)嵌套問(wèn)題5.(2023·云南保山·高三統(tǒng)考期末)定義域?yàn)镾KIPIF1<0的函數(shù)SKIPIF1<0,若關(guān)于SKIPIF1<0的方程SKIPIF1<0恰有5個(gè)不同的實(shí)數(shù)解SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,則所有實(shí)數(shù)SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0之和為(
)A.12 B.16 C.20 D.246.(2023·全國(guó)·高三福建省福州第八中學(xué)??计谀┒x在SKIPIF1<0上函數(shù)SKIPIF1<0,若關(guān)于SKIPIF1<0的方程SKIPIF1<0(其中SKIPIF1<0)有SKIPIF1<0個(gè)不同的實(shí)根SKIPIF1<0,SKIPIF1<0,…,SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<07.(2023·四川廣安·高三四川省鄰水縣第二中學(xué)校考階段練習(xí))設(shè)定義域?yàn)镽的函數(shù)SKIPIF1<0,若關(guān)于x的方程SKIPIF1<0有3個(gè)不同的實(shí)數(shù)解x1、x2、x3且x1<x2<x3,則下列說(shuō)法中錯(cuò)誤的是(
)A.SKIPIF1<0 B.1+a+b=0C.x1+x3=SKIPIF1<0 D.x1+x3>2x28.(2023·安徽亳州·高三安徽省亳州市第一中學(xué)校考階段練習(xí))設(shè)定義域?yàn)镽的函數(shù)SKIPIF1<0,若關(guān)于x的方程SKIPIF1<0有且僅有三個(gè)不同的實(shí)數(shù)解SKIPIF1<0,且SKIPIF1<0.下列說(shuō)法錯(cuò)誤的是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<09.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知函數(shù)SKIPIF1<0,若關(guān)于SKIPIF1<0的方程SKIPIF1<0恰有6個(gè)不同的實(shí)數(shù)解,則SKIPIF1<0的取值情況不可能的是(
)A.SKIPIF1<0,SKIPIF1<0 B.SKIPIF1<0,SKIPIF1<0C.SKIPIF1<0,SKIPIF1<0 D.SKIPIF1<0,SKIPIF1<003函數(shù)整數(shù)解問(wèn)題10.(2023·黑龍江綏化·高三校考階段練習(xí))已知函數(shù)SKIPIF1<0,若SKIPIF1<0有且只有兩個(gè)整數(shù)解,則k的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<011.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知不等式SKIPIF1<0的解集中僅有2個(gè)整數(shù),則實(shí)數(shù)k的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<012.(2023·全國(guó)·高三校聯(lián)考階段練習(xí))已知函數(shù)SKIPIF1<0,若對(duì)任意的SKIPIF1<0,都有SKIPIF1<0成立,則整數(shù)a的最大值為(
)A.3 B.4 C.5 D.613.(2023·江蘇蘇州·高三校考)已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)存在極值點(diǎn),且SKIPIF1<0在R上恰好有唯一整數(shù)解,則實(shí)數(shù)SKIPIF1<0的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<004唯一零點(diǎn)求值問(wèn)題14.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知函數(shù)SKIPIF1<0有唯一零點(diǎn),則負(fù)實(shí)數(shù)SKIPIF1<0()A.SKIPIF1<0 B.SKIPIF1<0 C.-3 D.-215.(2023·全國(guó)·高三階段練習(xí))已知函數(shù)SKIPIF1<0有唯一零點(diǎn),則SKIPIF1<0A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.116.(2023·云南曲靖·高三曲靖一中??茧A段練習(xí))已知函數(shù)SKIPIF1<0有唯一零點(diǎn),則SKIPIF1<0的值為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<017.(2023·山西·高三統(tǒng)考)已知數(shù)列SKIPIF1<0的首項(xiàng)SKIPIF1<0,函數(shù)SKIPIF1<0有唯一零點(diǎn),則通項(xiàng)SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<005等高線(xiàn)問(wèn)題18.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知函數(shù)SKIPIF1<0,其中SKIPIF1<0,若存在實(shí)數(shù)SKIPIF1<0,使得關(guān)于SKIPIF1<0的方程SKIPIF1<0恰有三個(gè)互異的實(shí)數(shù)解,則實(shí)數(shù)SKIPIF1<0的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<019.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知函數(shù)SKIPIF1<0,若關(guān)于SKIPIF1<0的方程SKIPIF1<0有四個(gè)不等根SKIPIF1<0,則SKIPIF1<0的值是(
)A.0 B.2 C.4 D.820.(2023·寧夏·高三寧夏大學(xué)附屬中學(xué)??茧A段練習(xí))已知函數(shù)SKIPIF1<0,若關(guān)于x的方程SKIPIF1<0有四個(gè)不同實(shí)數(shù)解SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0的取值范圍為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<021.(2023·湖北武漢·高一期末)已知函數(shù)SKIPIF1<0,若關(guān)于SKIPIF1<0的方程SKIPIF1<0有四個(gè)不同的實(shí)數(shù)解SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0的最小值為(
)A.SKIPIF1<0 B.8 C.SKIPIF1<0 D.SKIPIF1<006分段函數(shù)零點(diǎn)問(wèn)題22.(2023·四川綿陽(yáng)·四川省綿陽(yáng)南山中學(xué)??家荒#┮阎猄KIPIF1<0,函數(shù)SKIPIF1<0SKIPIF1<0,若SKIPIF1<0恰有2個(gè)零點(diǎn),則SKIPIF1<0的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<023.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知函數(shù)SKIPIF1<0若函數(shù)SKIPIF1<0有三個(gè)零點(diǎn),則(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<024.(2023·廣東廣州·高三廣州市真光中學(xué)校考期末)定義在SKIPIF1<0上的奇函數(shù)SKIPIF1<0,當(dāng)SKIPIF1<0時(shí),SKIPIF1<0,則關(guān)于SKIPIF1<0的函數(shù)SKIPIF1<0的所有零點(diǎn)之和為(
)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<025.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知函數(shù)SKIPIF1<0,則函數(shù)SKIPIF1<0的零點(diǎn)個(gè)數(shù)是(
)A.4 B.5 C.6 D.707函數(shù)對(duì)稱(chēng)問(wèn)題26.(2023·陜西渭南·高三渭南市瑞泉中學(xué)校考階段練習(xí))若直角坐標(biāo)平面內(nèi)SKIPIF1<0,SKIPIF1<0兩點(diǎn)滿(mǎn)足:①點(diǎn)SKIPIF1<0,SKIPIF1<0都在函數(shù)SKIPIF1<0的圖象上;②點(diǎn)SKIPIF1<0,SKIPIF1<0關(guān)于原點(diǎn)對(duì)稱(chēng),則稱(chēng)點(diǎn)SKIPIF1<0是函數(shù)SKIPIF1<0的一個(gè)“姊妹點(diǎn)對(duì)”點(diǎn)對(duì)SKIPIF1<0與SKIPIF1<0可看作是同一個(gè)“姊妹點(diǎn)對(duì)”.已知函數(shù)SKIPIF1<0恰有兩個(gè)“姊妹點(diǎn)對(duì)”,則實(shí)數(shù)SKIPIF1<0的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<027.(2023·湖南長(zhǎng)沙·高三長(zhǎng)沙市雅禮實(shí)驗(yàn)中學(xué)??奸_(kāi)學(xué)考試)若直角坐標(biāo)平面內(nèi)SKIPIF1<0兩點(diǎn)滿(mǎn)足條件:①點(diǎn)SKIPIF1<0都在SKIPIF1<0的圖像上;②點(diǎn)SKIPIF1<0關(guān)于原點(diǎn)對(duì)稱(chēng),則對(duì)稱(chēng)點(diǎn)對(duì)SKIPIF1<0是函數(shù)的一個(gè)“兄弟點(diǎn)對(duì)”(點(diǎn)對(duì)SKIPIF1<0與SKIPIF1<0可看作一個(gè)“兄弟點(diǎn)對(duì)”SKIPIF1<0.已知函數(shù)SKIPIF1<0,則SKIPIF1<0的“兄弟點(diǎn)對(duì)”的個(gè)數(shù)為(
)A.2 B.3 C.4 D.528.(2023·全國(guó)·高三專(zhuān)題練習(xí))若不同兩點(diǎn)SKIPIF1<0、SKIPIF1<0均在函數(shù)SKIPIF1<0的圖象上,且點(diǎn)SKIPIF1<0、SKIPIF1<0關(guān)于原點(diǎn)對(duì)稱(chēng),則稱(chēng)SKIPIF1<0是函數(shù)SKIPIF1<0的一個(gè)“匹配點(diǎn)對(duì)”(點(diǎn)對(duì)SKIPIF1<0與SKIPIF1<0視為同一個(gè)“匹配點(diǎn)對(duì)”).已知SKIPIF1<0恰有兩個(gè)“匹配點(diǎn)對(duì)”,則SKIPIF1<0的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<029.(2023·黑龍江哈爾濱·高一哈爾濱三中??计谥校┤艉瘮?shù)SKIPIF1<0圖象上存在不同的兩點(diǎn)SKIPIF1<0,SKIPIF1<0關(guān)于SKIPIF1<0軸對(duì)稱(chēng),則稱(chēng)點(diǎn)對(duì)SKIPIF1<0是函數(shù)SKIPIF1<0的一對(duì)“黃金點(diǎn)對(duì)”(注:點(diǎn)對(duì)SKIPIF1<0與SKIPIF1<0可看作同一對(duì)“黃金點(diǎn)對(duì)”).已知函數(shù)SKIPIF1<0則此函數(shù)的“黃金點(diǎn)對(duì)”有(
)A.0對(duì) B.1對(duì) C.2對(duì) D.3對(duì)08零點(diǎn)嵌套問(wèn)題30.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知函數(shù)SKIPIF1<0有三個(gè)不同的零點(diǎn)SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0的值為(
)A.1 B.3 C.4 D.931.(2023·重慶南岸·高三重慶市第十一中學(xué)校校考階段練習(xí))設(shè)定義在R上的函數(shù)SKIPIF1<0滿(mǎn)足SKIPIF1<0有三個(gè)不同的零點(diǎn)SKIPIF1<0且SKIPIF1<0則SKIPIF1<0的值是(
)A.81 B.-81 C.9 D.-932.(2023·江西宜春·高三江西省豐城中學(xué)??计谥校┮阎瘮?shù)SKIPIF1<0有三個(gè)不同的零點(diǎn)SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0的值為(
)A.3 B.6 C.9 D.3633.(2023·陜西·統(tǒng)考模擬預(yù)測(cè))已知函數(shù)SKIPIF1<0有三個(gè)不同的零點(diǎn)SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0的值為(
)A.3 B.4 C.9 D.1609函數(shù)零點(diǎn)問(wèn)題之三變量問(wèn)題34.(2023·全國(guó)·高二假期作業(yè))若存在兩個(gè)正實(shí)數(shù)SKIPIF1<0,SKIPIF1<0,使得等式SKIPIF1<0成立,其中SKIPIF1<0為自然對(duì)數(shù)的底數(shù),則實(shí)數(shù)SKIPIF1<0的取值范圍是A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<035.(2023·全國(guó)·高三專(zhuān)題練習(xí))若存在正實(shí)數(shù)x,y,使得等式SKIPIF1<0成立,其中e為自然對(duì)數(shù)的底數(shù),則a的取值范圍為(
)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<036.(2023·全國(guó)·高三專(zhuān)題練習(xí))若存在兩個(gè)正實(shí)數(shù)SKIPIF1<0,使得等式SKIPIF1<0成立,其中SKIPIF1<0為自然對(duì)數(shù)的底數(shù),則實(shí)數(shù)SKIPIF1<0的取值范圍為A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<037.(2023·全國(guó)·高三專(zhuān)題練習(xí))若存在兩個(gè)正實(shí)數(shù)SKIPIF1<0,SKIPIF1<0使等式SKIPIF1<0成立,其中SKIPIF1<0是自然對(duì)數(shù)的底數(shù),則實(shí)數(shù)SKIPIF1<0的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<038.(2023·江西新余·統(tǒng)考二模)若存在兩個(gè)正數(shù)SKIPIF1<0,使得不等式SKIPIF1<0成立,其中SKIPIF1<0,SKIPIF1<0為自然對(duì)數(shù)的底數(shù),則實(shí)數(shù)SKIPIF1<0的取值范圍為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<010倍值函數(shù)39.(2023·寧夏銀川·高一??计谥校┖瘮?shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0,若存在閉區(qū)間SKIPIF1<0,使得函數(shù)SKIPIF1<0同時(shí)滿(mǎn)足:(1)SKIPIF1<0在SKIPIF1<0內(nèi)是單調(diào)函數(shù);(2)SKIPIF1<0在SKIPIF1<0上的值域?yàn)镾KIPIF1<0,則稱(chēng)區(qū)間SKIPIF1<0為SKIPIF1<0的“SKIPIF1<0倍值區(qū)間”.下列函數(shù):①SKIPIF1<0;②SKIPIF1<0;③SKIPIF1<0;④SKIPIF1<0.其中存在“SKIPIF1<0倍值區(qū)間”的有(
)A.①③ B.②③ C.②④ D.①②③④40.(2023·安徽·高三統(tǒng)考期末)函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0,若存在閉區(qū)間SKIPIF1<0,使得函數(shù)SKIPIF1<0滿(mǎn)足:①SKIPIF1<0在SKIPIF1<0內(nèi)是單調(diào)函數(shù);②SKIPIF1<0在SKIPIF1<0上的值域?yàn)镾KIPIF1<0,則稱(chēng)區(qū)間SKIPIF1<0為SKIPIF1<0的“倍值區(qū)間”.下列函數(shù)中存在“倍值區(qū)間”的有①;②SKIPIF1<0;③;④A.①②③④ B.①②④ C.①③④ D.①③41.(2023·全國(guó)·高三專(zhuān)題練習(xí))函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0,對(duì)給定的正數(shù)SKIPIF1<0,若存在閉區(qū)間SKIPIF1<0,使得函數(shù)SKIPIF1<0滿(mǎn)足:①SKIPIF1<0在SKIPIF1<0內(nèi)是單調(diào)函數(shù);②SKIPIF1<0在SKIPIF1<0上的值域?yàn)镾KIPIF1<0,則稱(chēng)區(qū)間SKIPIF1<0為SKIPIF1<0的SKIPIF1<0級(jí)“理想?yún)^(qū)間”.下列結(jié)論錯(cuò)誤的是(
)A.函數(shù)SKIPIF1<0(SKIPIF1<0)存在1級(jí)“理想?yún)^(qū)間”B.函數(shù)SKIPIF1<0(SKIPIF1<0)不存在2級(jí)“理想?yún)^(qū)間”C.函數(shù)SKIPIF1<0(SKIPIF1<0)存在3級(jí)“理想?yún)^(qū)間”D.函數(shù)SKIPIF1<0,SKIPIF1<0不存在4級(jí)“理想?yún)^(qū)間”42.(2023·全國(guó)·高三專(zhuān)題練習(xí))設(shè)函數(shù)的定義域?yàn)镈,若滿(mǎn)足條件:存在SKIPIF1<0,使SKIPIF1<0在SKIPIF1<0上的值域?yàn)镾KIPIF1<0,則稱(chēng)SKIPIF1<0為“倍縮函數(shù)”.若函數(shù)SKIPIF1<0為“倍縮函數(shù)”,則實(shí)數(shù)t的取值范圍是A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<011函數(shù)不動(dòng)點(diǎn)問(wèn)題43.(2023·全國(guó)·高三專(zhuān)題練習(xí))設(shè)函數(shù)SKIPIF1<0,若曲線(xiàn)SKIPIF1<0上存在點(diǎn)SKIPIF1<0,SKIPIF1<0使得SKIPIF1<0成立,則實(shí)數(shù)SKIPIF1<0的取值范圍為(
)A.SKIPIF1<0,SKIPIF1<0 B.SKIPIF1<0,SKIPIF1<0 C.SKIPIF1<0,SKIPIF1<0 D.SKIPIF1<0,SKIPIF1<044.(2023·黑龍江齊齊哈爾·高一齊齊哈爾市恒昌中學(xué)校??计谥校┰跀?shù)學(xué)中,布勞威爾不動(dòng)點(diǎn)定理是拓?fù)鋵W(xué)里一個(gè)非常重要的不動(dòng)點(diǎn)定理,它可應(yīng)用到有限維空間,并構(gòu)成一般不動(dòng)點(diǎn)定理的基石.布勞威爾不動(dòng)點(diǎn)定理得名于荷蘭數(shù)學(xué)家魯伊茲?布勞威爾(L.E.J.Brouwer),簡(jiǎn)單的講就是對(duì)于滿(mǎn)足一定條件的圖象不間斷的函數(shù)f(x),存在一個(gè)點(diǎn)x0,使得f(x0)=x0,那么我們稱(chēng)該函數(shù)為“不動(dòng)點(diǎn)“函數(shù).下列為“不動(dòng)點(diǎn)”函數(shù)的是(
)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<045.(2023·山東菏澤·統(tǒng)考一模)定義在實(shí)數(shù)集SKIPIF1<0上的函數(shù)SKIPIF1<0,如果SKIPIF1<0,使得SKIPIF1<0,則稱(chēng)SKIPIF1<0為函數(shù)SKIPIF1<0的不動(dòng)點(diǎn).給定函數(shù)SKIPIF1<0,SKIPIF1<0,已知函數(shù)SKIPIF1<0,SKIPIF1<0,SKIPIF1<0在SKIPIF1<0上均存在唯一不動(dòng)點(diǎn),分別記為SKIPIF1<0,則(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<012函數(shù)的旋轉(zhuǎn)問(wèn)題46.(2023·上海浦東新·高三上海市建平中學(xué)??奸_(kāi)學(xué)考試)在平面直角坐標(biāo)系中,已知點(diǎn)SKIPIF1<0的坐標(biāo)為SKIPIF1<0,將點(diǎn)SKIPIF1<0繞原點(diǎn)按逆時(shí)針?lè)较蛐D(zhuǎn)角SKIPIF1<0得到點(diǎn)SKIPIF1<0,再將點(diǎn)SKIPIF1<0繞原點(diǎn)按逆時(shí)針?lè)较蛐D(zhuǎn)角SKIPIF1<0得到SKIPIF1<0,…,如此繼續(xù)下去,得到前10個(gè)點(diǎn)SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,…,SKIPIF1<0.若SKIPIF1<0是公差為SKIPIF1<0的等差數(shù)列,且點(diǎn)SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,…,SKIPIF1<0在同一函數(shù)圖像上,則角SKIPIF1<0的取值可以是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<047.(2023·上海浦東新·高二上海市實(shí)驗(yàn)學(xué)校??奸_(kāi)學(xué)考試)2021年第十屆中國(guó)花卉博覽會(huì)舉辦在即,其中,以“蝶戀花”為造型的世紀(jì)館引人矚目(如圖①),而美妙的蝴蝶輪廓不僅帶來(lái)生活中的賞心悅目,也展示了極致的數(shù)學(xué)美學(xué)世界.數(shù)學(xué)家曾借助三角函數(shù)得到了蝴蝶曲線(xiàn)的圖像,探究如下:如圖②,平面上有兩定點(diǎn)SKIPIF1<0,兩動(dòng)點(diǎn)SKIPIF1<0,且SKIPIF1<0繞點(diǎn)SKIPIF1<0逆時(shí)針旋轉(zhuǎn)到SKIPIF1<0所形成的角記為SKIPIF1<0,設(shè)函數(shù)SKIPIF1<0,其中SKIPIF1<0令SKIPIF1<0,作SKIPIF1<0,隨著SKIPIF1<0的變化,就得到了點(diǎn)SKIPIF1<0的軌跡,其形似“蝴蝶”,則以下4幅圖中,點(diǎn)SKIPIF1<0的軌跡(考慮蝴蝶的朝向)最有可能為(
)A. B.C. D.48.(2023·全國(guó)·高三專(zhuān)題練習(xí))將函數(shù)SKIPIF1<0的圖象繞點(diǎn)SKIPIF1<0逆時(shí)針旋轉(zhuǎn)SKIPIF1<0,得到曲線(xiàn)SKIPIF1<0,對(duì)于每一個(gè)旋轉(zhuǎn)角SKIPIF1<0,曲線(xiàn)SKIPIF1<0都是一個(gè)函數(shù)的圖象,則SKIPIF1<0最大時(shí)的正切值為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<013構(gòu)造函數(shù)解不等式49.(2023·江蘇·高二專(zhuān)題練習(xí))已知定義在R上的偶函數(shù)SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0,則關(guān)于x的不等式SKIPIF1<0的解集為(
)A.(4,+∞) B.(-∞,4) C.(-∞,3) D.(3,+∞)50.(2023·江蘇·高二專(zhuān)題練習(xí))函數(shù)SKIPIF1<0定義域?yàn)镽,導(dǎo)函數(shù)為SKIPIF1<0,SKIPIF1<0滿(mǎn)足下列條件:①任意SKIPIF1<0,SKIPIF1<0恒成立,②SKIPIF1<0時(shí),SKIPIF1<0恒成立,則關(guān)于t的不等式:SKIPIF1<0的解集為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<051.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù),當(dāng)SKIPIF1<0時(shí),SKIPIF1<0,則不等式SKIPIF1<0的解集為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<052.(2023·全國(guó)·高二專(zhuān)題練習(xí))已知定義域?yàn)镾KIPIF1<0的函數(shù)SKIPIF1<0滿(mǎn)足SKIPIF1<0(SKIPIF1<0為函數(shù)SKIPIF1<0的導(dǎo)函數(shù)),則不等式SKIPIF1<0的解集為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<053.(2023·全國(guó)·高二專(zhuān)題練習(xí))已知定義在R上的可導(dǎo)函數(shù)SKIPIF1<0的導(dǎo)函數(shù)為SKIPIF1<0,滿(mǎn)足SKIPIF1<0且SKIPIF1<0為偶函數(shù),SKIPIF1<0為奇函數(shù),若SKIPIF1<0,則不等式SKIPIF1<0的解集為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<054.(2023·遼寧沈陽(yáng)·高三沈陽(yáng)鐵路實(shí)驗(yàn)中學(xué)??计谥校┰O(shè)SKIPIF1<0為函數(shù)SKIPIF1<0的導(dǎo)函數(shù),已知SKIPIF1<0,則下列結(jié)論正確的是(
)A.SKIPIF1<0在SKIPIF1<0既有極大值又有極小值 B.SKIPIF1<0在SKIPIF1<0既無(wú)極大值又無(wú)極小值C.SKIPIF1<0在SKIPIF1<0上有極大值 D.SKIPIF1<0在SKIPIF1<0上有極小值14導(dǎo)數(shù)中的距離問(wèn)題55.(2023·重慶·重慶南開(kāi)中學(xué)??家荒#┤魧?duì)任意的實(shí)數(shù)SKIPIF1<0,函數(shù)SKIPIF1<0在SKIPIF1<0上是增函數(shù),則實(shí)數(shù)SKIPIF1<0的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<056.(2023·四川綿陽(yáng)·統(tǒng)考一模)若存在實(shí)數(shù)SKIPIF1<0,使得關(guān)于SKIPIF1<0的不等式SKIPIF1<0(其中SKIPIF1<0為自然對(duì)數(shù)的底數(shù))成立,則實(shí)數(shù)SKIPIF1<0的取值集合為()A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<057.(2023·高二單元測(cè)試)設(shè)點(diǎn)SKIPIF1<0為圓SKIPIF1<0上的任意一點(diǎn),點(diǎn)SKIPIF1<0SKIPIF1<0,則線(xiàn)段SKIPIF1<0長(zhǎng)度的最小值為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<058.(2023·重慶·高二校聯(lián)考階段練習(xí))若實(shí)數(shù)SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0滿(mǎn)足SKIPIF1<0且SKIPIF1<0(其中SKIPIF1<0,SKIPIF1<0,SKIPIF1<0是自然對(duì)數(shù)底數(shù)),則SKIPIF1<0最小值為A.SKIPIF1<0 B.5 C.SKIPIF1<0 D.1015導(dǎo)數(shù)的同構(gòu)思想59.(2023·河南·高三洛陽(yáng)市第一高級(jí)中學(xué)校聯(lián)考階段練習(xí))已知函數(shù)SKIPIF1<0有兩個(gè)不同的零點(diǎn),則實(shí)數(shù)a的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<060.(2023·全國(guó)·安陽(yáng)市第二中學(xué)校聯(lián)考模擬預(yù)測(cè))已知關(guān)于x的不等式SKIPIF1<0在SKIPIF1<0上恒成立,則正數(shù)m的最大值為(
)A.SKIPIF1<0 B.0 C.e D.161.(2023·河南洛陽(yáng)·高三新安縣第一高級(jí)中學(xué)校考開(kāi)學(xué)考試)已知函數(shù)SKIPIF1<0,若關(guān)于SKIPIF1<0的不等式SKIPIF1<0恒成立,則實(shí)數(shù)SKIPIF1<0的取值范圍(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<062.(2023·寧夏石嘴山·高三平羅中學(xué)??茧A段練習(xí))已知函數(shù)SKIPIF1<0,在區(qū)間SKIPIF1<0內(nèi)任取兩個(gè)實(shí)數(shù)SKIPIF1<0,SKIPIF1<0且SKIPIF1<0,若不等式SKIPIF1<0恒成立,則實(shí)數(shù)SKIPIF1<0的取值范圍為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<016不等式恒成立之分離參數(shù)、分離函數(shù)、放縮法63.(2023·陜西安康·統(tǒng)考二模)已知SKIPIF1<0恒成立,則λ的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<064.(2023·江西·高三校聯(lián)考開(kāi)學(xué)考試)已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上恒小于0,則實(shí)數(shù)SKIPIF1<0的取值集合是(
)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<065.(2023·遼寧盤(pán)錦·盤(pán)錦市高級(jí)中學(xué)校考一模)已知函數(shù)SKIPIF1<0,若對(duì)于任意的實(shí)數(shù)SKIPIF1<0恒有SKIPIF1<0,則實(shí)數(shù)SKIPIF1<0的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<066.(2023·全國(guó)·高三專(zhuān)題練習(xí))若存在SKIPIF1<0使對(duì)于任意SKIPIF1<0不等式SKIPIF1<0恒成立,則實(shí)數(shù)SKIPIF1<0的最小值為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<017三次函數(shù)問(wèn)題67.(2023·河南·統(tǒng)考三模)已知SKIPIF1<0SKIPIF1<0為三次函數(shù),其圖象如圖所示.若SKIPIF1<0有9個(gè)零點(diǎn),則SKIPIF1<0的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<068.(2023·全國(guó)·高三專(zhuān)題練習(xí))韋達(dá)是法國(guó)杰出的數(shù)學(xué)家,其貢獻(xiàn)之一是發(fā)現(xiàn)了多項(xiàng)式方程根與系數(shù)的關(guān)系,如:設(shè)一元三次方程SKIPIF1<0的3個(gè)實(shí)數(shù)根為SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.已知函數(shù)SKIPIF1<0,直線(xiàn)SKIPIF1<0與SKIPIF1<0的圖象相切于點(diǎn)SKIPIF1<0,且交SKIPIF1<0的圖象于另一點(diǎn)SKIPIF1<0,則(
)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<069.(2023·重慶渝中·重慶巴蜀中學(xué)??寄M預(yù)測(cè))為響應(yīng)國(guó)家精準(zhǔn)扶貧政策,某工作組要在村外一湖岸邊修建一段道路(如圖中虛線(xiàn)處),要求該道路與兩條直線(xiàn)道路平滑連接(注:兩直線(xiàn)道路:SKIPIF1<0,SKIPIF1<0分別與該曲線(xiàn)相切于SKIPIF1<0,SKIPIF1<0,已知該彎曲路段為三次函數(shù)圖象的一部分,則該解析式為(
).A.SKIPIF1<0B.SKIPIF1<0C.SKIPIF1<0D.SKIPIF1<070.(2023·上海浦東新·高三上海市實(shí)驗(yàn)學(xué)校??茧A段練習(xí))已知SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,若三次函數(shù)SKIPIF1<0有三個(gè)零點(diǎn)SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,且滿(mǎn)足SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<018切線(xiàn)條數(shù)、公切線(xiàn)、切線(xiàn)重合與垂直問(wèn)題71.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知曲線(xiàn)SKIPIF1<0與SKIPIF1<0的兩條公切線(xiàn)所成角的正切值為SKIPIF1<0,則SKIPIF1<0(
)A.2 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<072.(2023·全國(guó)·高二專(zhuān)題練習(xí))若過(guò)點(diǎn)SKIPIF1<0可以作曲線(xiàn)SKIPIF1<0的三條切線(xiàn),則()A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<073.(2023·全國(guó)·高二專(zhuān)題練習(xí))過(guò)直線(xiàn)SKIPIF1<0上一點(diǎn)SKIPIF1<0可以作曲線(xiàn)SKIPIF1<0的兩條切線(xiàn),則點(diǎn)SKIPIF1<0橫坐標(biāo)SKIPIF1<0的取值范圍為(
)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<074.(2023·全國(guó)·高三專(zhuān)題練習(xí))若函數(shù)SKIPIF1<0與SKIPIF1<0的圖象存在公共切線(xiàn),則實(shí)數(shù)a的最大值為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<075.(2023·全國(guó)·高三專(zhuān)題練習(xí))若函數(shù)SKIPIF1<0的圖象上存在兩個(gè)不同的點(diǎn)SKIPIF1<0,SKIPIF1<0,使得曲線(xiàn)SKIPIF1<0在這兩點(diǎn)處的切線(xiàn)重合,則稱(chēng)函數(shù)SKIPIF1<0為“自重合”函數(shù).下列函數(shù)中既是奇函數(shù)又是“自重合”函數(shù)的是(
)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<076.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知函數(shù)SKIPIF1<0圖象上存在兩條互相垂直的切線(xiàn),且SKIPIF1<0,則SKIPIF1<0的最大值為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<019任意存在性問(wèn)題77.(2023·全國(guó)·高三專(zhuān)題練習(xí))定義:設(shè)函數(shù)SKIPIF1<0在SKIPIF1<0上的導(dǎo)函數(shù)為SKIPIF1<0,若SKIPIF1<0在SKIPIF1<0上也存在導(dǎo)函數(shù),則稱(chēng)函數(shù)SKIPIF1<0在SKIPIF1<0上存在二階導(dǎo)函數(shù),簡(jiǎn)記為SKIPIF1<0.若在區(qū)間SKIPIF1<0上SKIPIF1<0,則稱(chēng)函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上為“凹函數(shù)”.已知SKIPIF1<0在區(qū)間SKIPIF1<0上為“凹函數(shù)”,則實(shí)數(shù)SKIPIF1<0的取值范圍為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<078.(2023·黑龍江·黑龍江實(shí)驗(yàn)中學(xué)??家荒#┮阎瘮?shù)SKIPIF1<0,若存在SKIPIF1<0使得關(guān)于SKIPIF1<0的不等式SKIPIF1<0成立,則實(shí)數(shù)SKIPIF1<0的取值范圍(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<079.(2023·天津武清·高三天津市武清區(qū)楊村第一中學(xué)校考開(kāi)學(xué)考試)已知函數(shù)SKIPIF1<0,若存在實(shí)數(shù)SKIPIF1<0(SKIPIF1<0且SKIPIF1<0),使得SKIPIF1<0成立,則實(shí)數(shù)SKIPIF1<0的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<020雙參數(shù)最值問(wèn)題80.(2023·湖北黃岡·黃岡中學(xué)校考模擬預(yù)測(cè))已知不等式SKIPIF1<0(SKIPIF1<0,且SKIPIF1<0)對(duì)任意實(shí)數(shù)SKIPIF1<0恒成立,則SKIPIF1<0的最大值為_(kāi)___________.81.(2023·全國(guó)·模擬預(yù)測(cè))已知SKIPIF1<0,SKIPIF1<0,SKIPIF1<0恒成立,則SKIPIF1<0的最大值為_(kāi)_____.82.(2023·江蘇揚(yáng)州·高三校聯(lián)考期末)已知關(guān)于SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0時(shí),關(guān)于SKIPIF1<0的不等式SKIPIF1<0恒成立,則SKIPIF1<0的最小值為_(kāi)_____.83.(2023·重慶九龍坡·高一重慶市楊家坪中學(xué)??茧A段練習(xí))已知SKIPIF1<0,若SKIPIF1<0時(shí),關(guān)于SKIPIF1<0的不等式SKIPIF1<0恒成立,則SKIPIF1<0的最小值為_(kāi)_______84.(2023·上海浦東新·高一上海市進(jìn)才中學(xué)??计谥校┮阎猄KIPIF1<0,且SKIPIF1<0,若對(duì)SKIPIF1<0,不等式SKIPIF1<0恒成立,則SKIPIF1<0的最大值為_(kāi)__.21切線(xiàn)斜率與割線(xiàn)斜率85.(2023·山東臨沂·高三??茧A段練習(xí))已知函數(shù)SKIPIF1<0是定義在R上的函數(shù),且SKIPIF1<0是奇函數(shù),SKIPIF1<0是偶函數(shù),SKIPIF1<0SKIPIF1<0,記SKIPIF1<0,若對(duì)于任意的SKIPIF1<0,都有SKIPIF1<0,則實(shí)數(shù)a的取值范圍為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<086.(2023·湖北武漢·高一湖北省水果湖高級(jí)中學(xué)??茧A段練習(xí))已知函數(shù)SKIPIF1<0?SKIPIF1<0是定義在SKIPIF1<0上的函數(shù),其中SKIPIF1<0是奇函數(shù),SKIPIF1<0是偶函數(shù),且SKIPIF1<0,若對(duì)于任意SKIPIF1<0,都有SKIPIF1<0,則實(shí)數(shù)SKIPIF1<0的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<087.(2023·湖北黃岡·高一校考期中)已知函數(shù)SKIPIF1<0是定義在R上的函數(shù),其中SKIPIF1<0是奇函數(shù),SKIPIF1<0是偶函數(shù),且SKIPIF1<0,若對(duì)于任意SKIPIF1<0,都有SKIPIF1<0,則實(shí)數(shù)a的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<088.(2023·河南信陽(yáng)·高三??茧A段練習(xí))已知函數(shù)SKIPIF1<0是定義在R上的函數(shù),其中SKIPIF1<0是奇函數(shù),SKIPIF1<0是偶函數(shù),且SKIPIF1<0,若對(duì)于任意SKIPIF1<0,都有SKIPIF1<0,則實(shí)數(shù)SKIPIF1<0的取值范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<022最大值的最小值問(wèn)題(平口單峰函數(shù)、鉛錘距離)89.(2023·高一課時(shí)練習(xí))已知函數(shù)SKIPIF1<0,當(dāng)SKIPIF1<0時(shí),設(shè)SKIPIF1<0的最大值為SKIPIF1<0,則SKIPIF1<0的最小值為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.190.(2023·全國(guó)·高三專(zhuān)題練習(xí))已知函數(shù)SKIPIF1<0,當(dāng)SKIPIF1<0時(shí),SKIPIF1<0的最大值為SKIPIF1<0,若SKIPIF1<0的最小值為4,則實(shí)數(shù)SKIPIF1<0的取值范圍為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<091.(2023·江西宜春·校聯(lián)考模擬預(yù)測(cè))已知函數(shù)SKIPIF1<0,且SKIPIF1<0,滿(mǎn)足SKIPIF1<0,當(dāng)SKIPIF1<0時(shí),設(shè)函數(shù)SKIPIF1<0的最大值為SKIPIF1<0,則SKIPIF1<0的最小值為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<092.(2023·高二課時(shí)練習(xí))設(shè)SKIPIF1<0(SKIPIF1<0),當(dāng)SKIPIF1<0時(shí),SKIPIF1<0的最大值為SKIPIF1<0,則SKIPIF1<0的最小值為(
)A.SKIPIF1<0 B.1 C.SKIPIF1<0 D.293.(2023·浙江杭州·校聯(lián)考二模)設(shè)函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的最大值SKIPIF1<0的最小值為4,則符合條件的SKIPIF1<0有(
)①x2+SKIPIF1<0②SKIPIF1<0③SKIPIF1<0A.①② B.②③ C.①②③ D.①③23兩邊夾問(wèn)題和零點(diǎn)相同問(wèn)題94.(2023·河北石家莊·高一??茧A段練習(xí))已知SKIPIF1<0滿(mǎn)足SKIPIF1<0,其中SKIPIF1<0是自然對(duì)數(shù)的底數(shù),則SKIPIF1<0的值為(
)A.SKIPIF1<0 B.1 C.SKIPIF1<0 D.SKIPIF1<095.(2023·全國(guó)·高三專(zhuān)題練習(xí))若不等式SKIPIF1<0對(duì)SKIPIF1<0恒成立,則SKIPIF1<0的值等于(
)A.SKIPIF1<0 B.SKIPIF1<0 C.1 D.296.(2023·浙江·高一期末)若不等式SKIPIF1<0對(duì)SKIPIF1<0上恒成立,則SKIPIF1<0(
)A.SKIPIF
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