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專題06基本不等式及應(yīng)用1、【2022年新高考2卷】若x,y滿足x2A.x+y≤1 B.x+y≥?2C.x2+【答案】BC【解析】因為ab≤a+b22≤a2+b22(a,b∈R),由x2+y由x2+y2?xy=1可變形為x因為x2+y2?xy=1變形可得x?y=43+23故選:BC.2、(2021年新高考1卷)已知SKIPIF1<0,SKIPIF1<0是橢圓SKIPIF1<0:SKIPIF1<0的兩個焦點,點SKIPIF1<0在SKIPIF1<0上,則SKIPIF1<0的最大值為()A.13 B.12 C.9 D.6【答案】C【解析】由題,SKIPIF1<0,則SKIPIF1<0,所以SKIPIF1<0(當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立).故選:C.3、(2020全國3文12)已知函數(shù)SKIPIF1<0,則()A.SKIPIF1<0的最小值為2 B.SKIPIF1<0的圖像關(guān)于SKIPIF1<0軸對稱C.SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對稱 D.SKIPIF1<0的圖像關(guān)于直線SKIPIF1<0對稱【答案】D【解析】由題意得SKIPIF1<0.對于A,當(dāng)SKIPIF1<0時,SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時取等號;當(dāng)SKIPIF1<0時,SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時取等號,所以A錯誤.對于B,SKIPIF1<0,所以SKIPIF1<0是奇函數(shù),圖象關(guān)于原點對稱,所以B錯誤.對于C,SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0,SKIPIF1<0的圖象不關(guān)于直線SKIPIF1<0對稱,所以C錯誤.對于D,SKIPIF1<0,SKIPIF1<0,所以SKIPIF1<0,SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對稱,所以D正確.故選D.4、(2020山東)已知SKIPIF1<0,SKIPIF1<0,且SKIPIF1<0,則 ()A.SKIPIF1<0B.SKIPIF1<0C.SKIPIF1<0D.SKIPIF1<0【答案】ABD【解析】對于A,SKIPIF1<0SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立,故A正確;對于B,SKIPIF1<0,所以SKIPIF1<0,故B正確;對于C,SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立,故C不正確;對于D,因為SKIPIF1<0,所以SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立,故D正確,故選:ABD.5、(2020上海13)下列不等式恒成立的是 ()A.SKIPIF1<0B.SKIPIF1<0C.SKIPIF1<0D.SKIPIF1<0【答案】B【解析】由基本不等式可知SKIPIF1<0,故A不正確;SKIPIF1<0,即SKIPIF1<0恒成立,故B正確;當(dāng)SKIPIF1<0時,不等式不成立,故C不正確;當(dāng)SKIPIF1<0時,不等式不成立,故D不正確,故選B.6、(2020江蘇12)已知SKIPIF1<0,則SKIPIF1<0的最小值是.【答案】SKIPIF1<0【解析】SKIPIF1<0,故SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0,即SKIPIF1<0,SKIPIF1<0時,取等號.∴SKIPIF1<0.7、(2020天津14)已知SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0的最小值為_________.【答案】4【解析】SKIPIF1<0,SKIPIF1<0,SKIPIF1<0SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0=4時取等號,結(jié)合SKIPIF1<0,解得SKIPIF1<0,或SKIPIF1<0時,等號成立,故答案為:SKIPIF1<0.8、(2019天津理13)設(shè),則的最小值為.【答案】【解析】,,,則;

由基本不等式,(當(dāng)且僅當(dāng)時,即,且時,即或時,等號成立).故的最小值為.題組一運用基本不等式比較大小1-1、(2023·云南玉溪·統(tǒng)考一模)(多選題)已知SKIPIF1<0,且SKIPIF1<0則下列結(jié)論一定正確的有(

)A.SKIPIF1<0 B.SKIPIF1<0C.a(chǎn)b有最大值4 D.SKIPIF1<0有最小值9【答案】AC【分析】A、C選項,分別根據(jù)基本不等式計算即可得到;B選項找出反例即可;D選項由基本不等式“1”的代換計算,漏除了4.【詳解】A選項,SKIPIF1<0,A正確;B選項,找反例,當(dāng)SKIPIF1<0時,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,B不正確;C選項,SKIPIF1<0,SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時取“=”,C正確;D選項,SKIPIF1<0,D不正確.故選:AC.1-2、(2023·山西·統(tǒng)考一模)(多選題)設(shè)SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,則下列結(jié)論正確的是(

)A.SKIPIF1<0的最大值為SKIPIF1<0 B.SKIPIF1<0的最小值為SKIPIF1<0C.SKIPIF1<0的最小值為9 D.SKIPIF1<0的最小值為SKIPIF1<0【答案】ABC【分析】對于AD,利用基本不等式判斷即可;對于B,利用不等式SKIPIF1<0判斷即可,對于C,利用基本不等式“1”的妙用判斷即可.【詳解】對于A,因為SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時取等號,故A正確;對于B,因為SKIPIF1<0,故SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時取等號,即SKIPIF1<0的最小值SKIPIF1<0,故B正確;對于C,SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0且SKIPIF1<0,即SKIPIF1<0,SKIPIF1<0時取等號,所以SKIPIF1<0的最小值為9,故C正確;對于D,SKIPIF1<0,故SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時取等號,即SKIPIF1<0的最大值SKIPIF1<0,故D錯誤.故選:ABC.1-3、(2023·安徽宿州·統(tǒng)考一模)(多選題)已知SKIPIF1<0,且SKIPIF1<0,則下列不等關(guān)系成立的是(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0【答案】ABC【分析】利用基本不等式易知選項AB正確;利用對數(shù)運算法則和重要不等式可知C正確;將不等式SKIPIF1<0化簡整理可得SKIPIF1<0,構(gòu)造函數(shù)SKIPIF1<0利用函數(shù)單調(diào)性即可證明D錯誤.【詳解】由基本不等式可知,SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立,即A正確;易知SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立,即B正確;由重要不等式和對數(shù)運算法則可得:SKIPIF1<0,當(dāng)且僅當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立,即C正確;由SKIPIF1<0可得SKIPIF1<0,所以SKIPIF1<0,若SKIPIF1<0,即證明SKIPIF1<0,即SKIPIF1<0即需證明SKIPIF1<0,令函數(shù)SKIPIF1<0,則SKIPIF1<0,當(dāng)SKIPIF1<0時,SKIPIF1<0,即SKIPIF1<0在SKIPIF1<0上單調(diào)遞增,所以SKIPIF1<0時,解不等式SKIPIF1<0可得SKIPIF1<0即可,即SKIPIF1<0時不等式SKIPIF1<0成立;當(dāng)SKIPIF1<0時,SKIPIF1<0,即SKIPIF1<0在SKIPIF1<0上單調(diào)遞減,解不等式SKIPIF1<0可得SKIPIF1<0,即SKIPIF1<0時不等式SKIPIF1<0才成立;綜上可知,當(dāng)SKIPIF1<0時,不等式SKIPIF1<0才成立,所以D錯誤.故選:ABC.1-4、(2022·山東德州·高三期末)(多選題)已知SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,則下列結(jié)論正確的是()A.SKIPIF1<0的最小值為SKIPIF1<0 B.SKIPIF1<0的最小值為16C.SKIPIF1<0的最大值為SKIPIF1<0 D.SKIPIF1<0的最小值為SKIPIF1<0【答案】ACD【解析】由SKIPIF1<0可得,SKIPIF1<0,SKIPIF1<0(當(dāng)且僅當(dāng)SKIPIF1<0時,取等號),故A正確;SKIPIF1<0(當(dāng)且僅當(dāng)SKIPIF1<0時,取等號),即SKIPIF1<0,故D正確;SKIPIF1<0(當(dāng)且僅當(dāng)SKIPIF1<0時,取等號),SKIPIF1<0(當(dāng)且僅當(dāng)SKIPIF1<0時,取等號),即SKIPIF1<0,故B錯誤;SKIPIF1<0,即SKIPIF1<0(當(dāng)且僅當(dāng)SKIPIF1<0時,取等號),故C正確;故選:ACD題組二運用基本不等式求函數(shù)最值2-1、(2022·江蘇揚州·高三期末)已知正實數(shù)x,y滿足x+y=1,則SKIPIF1<0的最小值為__________.【答案】SKIPIF1<0【解析】由題意可知,SKIPIF1<0=SKIPIF1<0=SKIPIF1<0=SKIPIF1<0+SKIPIF1<0=(SKIPIF1<0+SKIPIF1<0)(x+y)=4+5+SKIPIF1<0+SKIPIF1<0≥9+2SKIPIF1<0=SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0=SKIPIF1<0,SKIPIF1<0時取等號,此時SKIPIF1<0,故SKIPIF1<0的最小值為SKIPIF1<0.故答案為:SKIPIF1<02-2、(2023·天津濱海新·統(tǒng)考三模)已知SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0的最小值為(

)A.4 B.6 C.8 D.10【答案】B【詳解】由SKIPIF1<0知SKIPIF1<0,結(jié)合SKIPIF1<0,以及換底公式可知,SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0,當(dāng)且僅當(dāng),SKIPIF1<0,即SKIPIF1<0時等號成立,即SKIPIF1<0時等號成立,故SKIPIF1<0的最小值為SKIPIF1<0,故選:B.2-3、(2023·浙江·統(tǒng)考模擬預(yù)測)已知正實數(shù)SKIPIF1<0滿足SKIPIF1<0,則SKIPIF1<0的最小值為(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0【答案】C【詳解】由題可得,SKIPIF1<0,則SKIPIF1<0,所以SKIPIF1<0SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0,即SKIPIF1<0時,取得等號,故選:C.2-4、(2023·遼寧葫蘆島·統(tǒng)考二模)若SKIPIF1<0,則SKIPIF1<0的最小值是(

)A.SKIPIF1<0 B.1C.2 D.SKIPIF1<0【答案】C【詳解】SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時取等號,因此SKIPIF1<0,即SKIPIF1<0,解得SKIPIF1<0,所以當(dāng)SKIPIF1<0時,SKIPIF1<0取得最小值2.故選:C2-5、(2022年重慶市高三月考試卷)已知SKIPIF1<0,則SKIPIF1<0的最小值是______.【答案】SKIPIF1<0【解析】【詳解】由于SKIPIF1<0,SKIPIF1<0,所以SKIPIF1<0,SKIPIF1<0SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時等號成立.故答案為:SKIPIF1<0題組三運用基本不等式處理多元問題3-1、【2022·廣東省陽春市第一中學(xué)10月月考】已知不等式SKIPIF1<0的解集為SKIPIF1<0,則SKIPIF1<0__________,SKIPIF1<0的最小值為__________.【答案】①.SKIPIF1<0②.8【解析】由題知SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0,即SKIPIF1<0時取等號,故SKIPIF1<0的最小值為8.故答案為:SKIPIF1<0;SKIPIF1<03-2、(2022·湖北省仙桃中學(xué)模擬預(yù)測)已知SKIPIF1<0,則SKIPIF1<0的最小值為______________________

.【答案】1【分析】先根據(jù)解SKIPIF1<0,利用基本不等式求得SKIPIF1<0的最小值為9,再由SKIPIF1<0SKIPIF1<0,利用對勾函數(shù)的性質(zhì)求解.【詳解】解:因為SKIPIF1<0,所以SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0,即SKIPIF1<0時,等號成立,所以SKIPIF1<0,SKIPIF1<0,令SKIPIF1<0,因為SKIPIF1<0在SKIPIF1<0上遞增,所以SKIPIF1<0,故答案為:13-3、(2022·江蘇南通如東縣期中)已知a>0,b>0,c>0,eqa\s\up6(2)-ab+9b\s\up6(2)-5c=0,當(dāng)eq\f(c,ab)最小時,eqx\s\up6(2)-3x≥a+b-\f(1,3)c恒成立,則x的取值集合是▲.【答案】{x|x≤-1或x≥4}【解析】由題意可知a>0,b>0,c>0,a2-ab+9b2-5c=0,等式兩邊同除ab,可得eq\f(a,b)-1+eq\f(9b,a)=eq\f(5c,ab),所以eq\f(a,b)-1+eq\f(9b,a)≥2EQ\R(,\F(a,b)·\F(9b,a))-1=5,(當(dāng)且僅當(dāng)eq\f(a,b)=eq\f(9b,a)時等號成立),故eq\f(c,ab)的最小值為1(a=3b),所以c=ab=3b2,則a+b-eq\f(1,3)c=4b-b2,所以a+b-eq\f(1,3)c的最大值為4,故x2-3x≥4,解得x≤-1或x≥4題組四不等式的綜合運用4-1、(2023·安徽淮北·統(tǒng)考一模)(多選題)已知SKIPIF1<0是SKIPIF1<0的邊SKIPIF1<0上的一點(不包含頂點),且SKIPIF1<0,則(

)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<0【答案】AD【分析】利用平面向量的線性運算,結(jié)合基本不等式,驗證各選項的結(jié)果.【詳解】SKIPIF1<0是SKIPIF1<0的邊SKIPIF1<0上的一點(不包含頂點),則有SKIPIF1<0,得SKIPIF1<0,即SKIPIF1<0,又SKIPIF1<0,∴SKIPIF1<0,可得SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,所以A選項正確,B選項錯誤;SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時等號成立,所以SKIPIF1<0,C選項錯誤;SKIPIF1<0,D選項正確.故選:AD.4-2、(2022·湖北·荊門市龍泉中學(xué)二模)正項等比數(shù)列SKIPIF1<0中,SKIPIF1<0成等差數(shù)列,且存在兩項SKIPIF1<0使得SKIPIF1<0,則SKIPIF1<0的最小值是(

)A.2 B.SKIPIF1<0 C.SKIPIF1<0 D.不存在【答案】B【分析】由等比數(shù)列通項公式及等差中項的性質(zhì)可得SKIPIF1<0,進而有SKIPIF1<0,利用基本不等式“1”的代換求目標(biāo)式最小值,注意等號是否成立.【詳解】由題設(shè)SKIPIF1<0,若SKIPIF1<0公比為SKIPIF1<0且SKIPIF1<0,則SKIPIF1<0,所以SKIPIF1<0,由SKIPIF1<0,則SKIPIF1<0,故SKIPIF1<0,可得SKIPIF1<0,所以SKIPIF1<0,而SKIPIF1<0,故等號不成立,又SKIPIF1<0,故當(dāng)SKIPIF1<0時SKIPIF1<0,當(dāng)SKIPIF1<0時SKIPIF1<0,顯然SKIPIF1<0,故SKIPIF1<0時SKIPIF1<0最小值為SKIPIF1<0.故選:B4-3、(2022·山東師范大學(xué)附中模擬預(yù)測)已知隨機變量SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0的最小值為________.【答案】SKIPIF1<0【分析】先由正態(tài)分布對稱性求出SKIPIF1<0,進而利用基本不等式“1”的妙用求解最小值.【詳解】由正態(tài)分布的對稱性可知:SKIPIF1<0,解得:SKIPIF1<0,因為SKIPIF1<0,所以SKIPIF1<0,由基本不等式得:SKIPIF1<0SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0,即SKIPIF1<0時等號成立,所以不等式得最小值為SKIPIF1<0故答案為:SKIPIF1<04-4、(2022·河北保定·一模)(多選題)下面描述正確的是(

)A.已知SKIPIF1<0,SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0B.函數(shù)SKIPIF1<0,若SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0的最小值是SKIPIF1<0C.已知SKIPIF1<0,則SKIPIF1<0的最小值為SKIPIF1<0D.已知SKIPIF1<0,則SKIPIF1<0的最小值為SKIPIF1<0【答案】AC【分析】對于選項A,利用基本不等式結(jié)合對數(shù)運算求解判斷;對于選項B:結(jié)合對數(shù)的性質(zhì),利用對勾函數(shù)的單調(diào)性求解判斷;C,用“1”的代換,利用基本不等式求解判斷;對于選項D,將SKIPIF1<0,轉(zhuǎn)化為SKIPIF1<0,利用二次函數(shù)的性質(zhì)求解判斷.【詳解】對于選項A,∵SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,∴SKIPIF1<0,∴SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時取等號,∴SKIPIF1<0,∴A正確;對于選項B:因為SKIPIF1<0,所以SKIPIF1<0,又SKIPIF1<0,所以由對勾函數(shù)的單調(diào)性可知函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞減,所以SKIPIF1<0,即SKIPIF1<0,故B不正確;對于選項C,根據(jù)題意,已知SKIPIF1<0,則SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0,即SKIPIF1<0時,等號成立,所以SKIPIF1<0,故C正確;對于選項D,SKIPIF1<0,令SKIPIF1<0,所以SKIPIF1<0,所以SKIPIF1<0,此時SKIPIF1<0無解,所以選項D不正確,故選:AC.1、(2022·山東棗莊·高三期末)已知SKIPIF1<0,則SKIPIF1<0的最小值是().A.6 B.5 C.4 D.3【答案】D【解析】SKIPIF1<0,故SKIPIF1<0,SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立,故SKIPIF1<0的最小值是3.故選:D.2、(2022年遼寧撫順市高三月考模擬試卷)對任意的正實數(shù)SKIPIF1<0,SKIPIF1<0,SKIPIF1<0恒成立,則SKIPIF1<0的最小值為()A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0【答案】B【解析】【詳解】依題意得SKIPIF1<0恒成立,因為SKIPIF1<0,SKIPIF1<0,所以SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立,所以SKIPIF1<0,即SKIPIF1<0的最小值為SKIPIF1<0.故選:B3、(2023·山東煙臺·統(tǒng)考三模)(多選題)已知SKIPIF1<0且SKIPIF1<0,則(

)A.SKIPIF1<0的最大值為SKIPIF1<0 B.SKIPIF1<0的最大值為2C.SKIPIF1<0的最小值為6 D.SKIPIF1<0的最小值為4【答案】BC【詳解】對于A,因為SKIPIF1<0,所以SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立,故SKIPIF1<0錯誤;對于B,因為SKIPIF1<0,所以SKIPIF1<0,即SKIPIF1<0,SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立,故B正確;對于C,由SKIPIF1<0得SKIPIF1<0,所以SKIPIF1<0,因為SKIPIF1<0,所以SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立,故C正確;對于D,令SKIPIF1<0,則SKIPIF1<0,所以SKIPIF1<0的最小值不是4,D錯誤.故選:BC.4、(2023·重慶·統(tǒng)考三模)(多選題)已知SKIPIF1<0,SKIPIF1<0,且SKIPIF1<0,則下列結(jié)論正確的是(

)A.SKIPIF1<0的取值范圍是SKIPIF1<0 B.SKIPIF1<0的取值范圍是SKIPIF1<0C.SKIPIF1<0的最小值是SKIPIF1<0 D.SKIPIF1<0的最小值是3【答案】BC【詳解】對于A,因為SKIPIF1<0,SKIPIF1<0,所以SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時取等號,由SKIPIF1<0,即SKIPIF1<0,解得SKIPIF1<0,即SKIPIF1<0,A錯誤;對于B,由SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時取等號,得SKIPIF1<0,所以SKIPIF1<0,又SKIPIF1<0,所以SKIPIF1<0,即SKIPIF1<0,故B正確;對C選項,因為SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,得SKIPIF1<0,所以SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0,即SKIPIF1<0時等號成立,C正確,對于D,C選項知:SKIPIF1<0,則SKIPIF1<0SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0,即SKIPIF1<0時等號成立,但SKIPIF1<0,所以SKIPIF1<0.(等號取不到),故D錯誤;故選:BC.5、(2023·山東煙臺·統(tǒng)考三模)(多選題)已知SKIPIF1<0且SKIPIF1<0,則(

)A.SKIPIF1<0的最大值為SKIPIF1<0 B.SKIPIF1<0的最大值為2C.SKIPIF1<0的最小值為6 D.SKIPIF1<0的最小值為4【答案】BC【詳解】對于A,因為SKIPIF1<0,所以SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立,故SKIPIF1<0錯誤;對于B,因為SKIPIF1<0,所以SKIPIF1<0,即SKIPIF1<0,SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時,等號成立,故B正確;對于C,由SKIPIF

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