人教A版高中數(shù)學（選擇性必修二）同步講義第12講 5.3.1函數(shù)的單調(diào)性（原卷版）

第03講5.3.1函數(shù)的單調(diào)性課程標準學習目標①理解導數(shù)與函數(shù)的單調(diào)性的關(guān)系。②掌握利用導數(shù)判斷函數(shù)單調(diào)性的方法。③能利用導數(shù)求不超過三次多項式函數(shù)的單調(diào)區(qū)間。④會利用導數(shù)證明一些簡單的不等式問題。⑤掌握利用導數(shù)研究含參數(shù)的單調(diào)性的基本方法。通過本節(jié)課要求能利用函數(shù)的導數(shù)判斷函數(shù)的單調(diào)性，會求簡單函數(shù)的單調(diào)區(qū)間，能證明簡單的不等式，會利用導數(shù)解決單調(diào)性與含參數(shù)相關(guān)的問題.知識點01：函數(shù)的單調(diào)性與導數(shù)的關(guān)系（導函數(shù)看正負，原函數(shù)看增減）函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)可導，(1)若SKIPIF1<0，則SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)是單調(diào)遞增函數(shù)；(2)若SKIPIF1<0，則SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)是單調(diào)遞減函數(shù)；(3)若恒有SKIPIF1<0，則SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)是常數(shù)函數(shù)．注意：討論函數(shù)的單調(diào)性或求函數(shù)的單調(diào)區(qū)間的實質(zhì)是解不等式，求解時，要堅持“定義域優(yōu)先”原則條件恒有結(jié)論函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上可導SKIPIF1<0SKIPIF1<0在SKIPIF1<0內(nèi)單調(diào)遞增SKIPIF1<0SKIPIF1<0在SKIPIF1<0內(nèi)單調(diào)遞減SKIPIF1<0SKIPIF1<0在SKIPIF1<0內(nèi)是常數(shù)函數(shù)【即學即練1】（2023下·新疆巴音郭楞·高二校考期末）如圖所示是函數(shù)SKIPIF1<0的導函數(shù)SKIPIF1<0的圖象，則下列判斷中正確的是（

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A．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上是減函數(shù)B．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上是減函數(shù)C．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上是減函數(shù)D．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上是增函數(shù)【答案】A【詳解】對于選項A：當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0在SKIPIF1<0上單調(diào)遞減，故A正確；對于選項B：當SKIPIF1<0時，SKIPIF1<0；當SKIPIF1<0時，SKIPIF1<0；則SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，在SKIPIF1<0上單調(diào)遞減，故B錯誤；對于選項C：當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，故C錯誤；對于選項D：當SKIPIF1<0時，SKIPIF1<0，則SKIPIF1<0在SKIPIF1<0上單調(diào)遞減，故D錯誤；故選：A.知識點02：求已知函數(shù)（不含參）的單調(diào)區(qū)間①求SKIPIF1<0的定義域②求SKIPIF1<0③令SKIPIF1<0，解不等式，求單調(diào)增區(qū)間④令SKIPIF1<0，解不等式，求單調(diào)減區(qū)間注：求單調(diào)區(qū)間時，令SKIPIF1<0（或SKIPIF1<0）不跟等號.【即學即練2】（2023下·四川資陽·高二統(tǒng)考期末）函數(shù)SKIPIF1<0的單調(diào)遞減區(qū)間為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0，SKIPIF1<0【答案】A【詳解】因為SKIPIF1<0，所以函數(shù)SKIPIF1<0的定義域為SKIPIF1<0，所以SKIPIF1<0，由SKIPIF1<0有：SKIPIF1<0，所以函數(shù)SKIPIF1<0的單調(diào)遞減區(qū)間為SKIPIF1<0，故B，C，D錯誤.故選：A.知識點03：由函數(shù)SKIPIF1<0的單調(diào)性求參數(shù)的取值范圍的方法1、已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)①已知SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞增SKIPIF1<0SKIPIF1<0，SKIPIF1<0恒成立.②已知SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞減SKIPIF1<0SKIPIF1<0，SKIPIF1<0恒成立.注：已知單調(diào)性，等價條件中的不等式含等號.2、已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上存在單調(diào)區(qū)間①已知SKIPIF1<0在區(qū)間SKIPIF1<0上存在單調(diào)增區(qū)間SKIPIF1<0SKIPIF1<0使得SKIPIF1<0有解②已知SKIPIF1<0在區(qū)間SKIPIF1<0上存在單調(diào)減區(qū)間SKIPIF1<0SKIPIF1<0使得SKIPIF1<0有解3、已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上不單調(diào)SKIPIF1<0SKIPIF1<0，使得SKIPIF1<0有變號零點【即學即練3】（2023上·新疆·高三校聯(lián)考期中）已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞增，則SKIPIF1<0的取值范圍為.【答案】SKIPIF1<0【詳解】因為SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞增，所以當SKIPIF1<0時，SKIPIF1<0恒成立，即SKIPIF1<0在SKIPIF1<0恒成立，又SKIPIF1<0，所以SKIPIF1<0.故答案為：SKIPIF1<0.【即學即練4】（2023上·貴州貴陽·高三清華中學校考階段練習）已知函數(shù)SKIPIF1<0存在單調(diào)遞減區(qū)間，則實數(shù)SKIPIF1<0的取值范圍是.【答案】SKIPIF1<0【詳解】函數(shù)SKIPIF1<0的定義域為SKIPIF1<0，求導得SKIPIF1<0，依題意，不等式SKIPIF1<0在SKIPIF1<0上有解，等價于SKIPIF1<0在SKIPIF1<0上有解，而SKIPIF1<0，當且僅當SKIPIF1<0時取等號，則SKIPIF1<0，所以實數(shù)a的取值范圍是SKIPIF1<0.故答案為：SKIPIF1<0．知識點04：含參問題討論單調(diào)性第一步：求SKIPIF1<0的定義域第二步：求SKIPIF1<0（導函數(shù)中有分母通分）第三步：確定導函數(shù)有效部分，記為SKIPIF1<0對于SKIPIF1<0進行求導得到SKIPIF1<0，對SKIPIF1<0初步處理（如通分），提出SKIPIF1<0的恒正部分，將該部分省略，留下的部分則為SKIPIF1<0的有效部分（如：SKIPIF1<0，則記SKIPIF1<0為SKIPIF1<0的有效部分）.接下來就只需考慮導函數(shù)有效部分，只有該部分決定SKIPIF1<0的正負.第四步：確定導函數(shù)有效部分SKIPIF1<0的類型：①SKIPIF1<0為一次型（或可化為一次型）②SKIPIF1<0為二次型（或可化為二次型）第五步：通過分析導函數(shù)有效部分，討論SKIPIF1<0的單調(diào)性題型01求函數(shù)的單調(diào)區(qū)間【典例1】（2022下·湖北·高二統(tǒng)考期末）函數(shù)SKIPIF1<0的單調(diào)遞減區(qū)間為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023下·河北滄州·高二校考階段練習）函數(shù)SKIPIF1<0的單調(diào)遞減區(qū)間是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式1】（多選）（2023下·吉林長春·高二長春外國語學校?？计谥校┖瘮?shù)SKIPIF1<0的一個單調(diào)遞增區(qū)間是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0題型02函數(shù)與導函數(shù)圖象間的關(guān)系【典例1】（2023·高二課時練習）已知函數(shù)SKIPIF1<0的導函數(shù)SKIPIF1<0的圖象如圖，則下列結(jié)論正確的是（

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A．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞增B．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞減C．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞增D．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞增【典例2】（2022下·廣東深圳·高二統(tǒng)考期末）設(shè)SKIPIF1<0是函數(shù)SKIPIF1<0的導函數(shù)，SKIPIF1<0的圖象如圖所示，則SKIPIF1<0的圖象最有可能的是（

）A． B．C． D．【變式1】（2023下·四川成都·高二四川省成都市新都一中校聯(lián)考期中）已知函數(shù)SKIPIF1<0的導函數(shù)SKIPIF1<0的圖像如圖所示，則函數(shù)SKIPIF1<0（

）A．在SKIPIF1<0上單調(diào)遞增 B．在SKIPIF1<0上單調(diào)遞減C．在SKIPIF1<0上單調(diào)遞增 D．在SKIPIF1<0上單調(diào)遞減【變式2】（2022·湖南·校聯(lián)考二模）設(shè)函數(shù)SKIPIF1<0在定義域內(nèi)可導，SKIPIF1<0的圖象如圖所示，則其導函數(shù)SKIPIF1<0的圖象可能是（

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A． B．C． D．題型03已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)，求參數(shù)【典例1】（2023上·廣西·高三南寧三中校聯(lián)考階段練習）若函數(shù)SKIPIF1<0是SKIPIF1<0上的減函數(shù)，則實數(shù)SKIPIF1<0的最大值為．【典例2】（2023·海南·校聯(lián)考模擬預測）設(shè)SKIPIF1<0且SKIPIF1<0，若函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，則a的取值范圍是．【典例3】（2023上·遼寧大連·高三大連市金州高級中學校考期中）若函數(shù)SKIPIF1<0在SKIPIF1<0具有單調(diào)性，則a的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式1】（2023上·江蘇蘇州·高三常熟中學校考階段練習）已知函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，則實數(shù)a的取值范圍是．【變式2】（2023·海南省直轄縣級單位·校考模擬預測）若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞增，則實數(shù)SKIPIF1<0的取值范圍為．題型04已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上存在單調(diào)區(qū)間，求參數(shù)【典例1】（2023上·浙江寧波·高二鎮(zhèn)海中學?？计谥校┤艉瘮?shù)SKIPIF1<0在區(qū)間SKIPIF1<0上有單調(diào)遞增區(qū)間，則實數(shù)SKIPIF1<0的取值范圍是．【典例2】（2023下·江西撫州·高二江西省臨川第二中學?？茧A段練習）函數(shù)SKIPIF1<0在SKIPIF1<0上存在單調(diào)遞增區(qū)間，則SKIPIF1<0的取值范圍是.【變式1】（2023下·廣西·高二校聯(lián)考期中）若函數(shù)SKIPIF1<0在SKIPIF1<0存在單調(diào)遞減區(qū)間，則a的取值范圍為．題型05已知函數(shù)SKIPIF1<0在的單調(diào)區(qū)間為（是）SKIPIF1<0，求參數(shù)【典例1】（2023下·高二課時練習）已知函數(shù)SKIPIF1<0的單調(diào)遞減區(qū)間是SKIPIF1<0，則SKIPIF1<0.題型06已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上不單調(diào)，求參數(shù)【典例1】（2023下·湖北·高二校聯(lián)考階段練習）若函數(shù)SKIPIF1<0在其定義域的一個子區(qū)間SKIPIF1<0內(nèi)不是單調(diào)函數(shù)，則實數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2022上·河南·高三校聯(lián)考階段練習）已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上不是單調(diào)函數(shù)，則實數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式1】（2022·全國·高二專題練習）已知函數(shù)SKIPIF1<0．若SKIPIF1<0在SKIPIF1<0內(nèi)不單調(diào)，則實數(shù)a的取值范圍是．【變式2】（2022下·福建漳州·高二福建省漳州第一中學校考階段練習）若函數(shù)SKIPIF1<0在區(qū)間(1，4)上不單調(diào)，則實數(shù)a的取值范圍是.題型07含參問題討論單調(diào)性(導函數(shù)有效部分是一次型)【典例1】（2023上·陜西咸陽·高三統(tǒng)考期中）已知函數(shù)SKIPIF1<0.(1)若SKIPIF1<0，求函數(shù)SKIPIF1<0的極值；(2)求函數(shù)SKIPIF1<0的單調(diào)區(qū)間.【典例2】（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0．求函數(shù)SKIPIF1<0的單調(diào)區(qū)間．【變式1】（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0，討論函數(shù)SKIPIF1<0的單調(diào)性．【變式2】（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0．討論函數(shù)SKIPIF1<0的單調(diào)性．題型08含參問題討論單調(diào)性(導函數(shù)有效部分是二次型且可因式分解)【典例1】（2023上·江蘇揚州·高三儀征市第二中學?？计谥校┮阎瘮?shù)SKIPIF1<0，其中SKIPIF1<0．(1)若SKIPIF1<0是函數(shù)SKIPIF1<0的極值點，求a的值；(2)若SKIPIF1<0，討論函數(shù)SKIPIF1<0的單調(diào)性．【典例2】（2024·全國·高三專題練習）已知函數(shù)SKIPIF1<0，其中SKIPIF1<0.求函數(shù)SKIPIF1<0的單調(diào)區(qū)間；【典例3】（2023上·甘肅慶陽·高三?？茧A段練習）已知函數(shù)SKIPIF1<0SKIPIF1<0(1)若SKIPIF1<0，求曲線SKIPIF1<0在點SKIPIF1<0處的切線方程；(2)當SKIPIF1<0時，求函數(shù)SKIPIF1<0的單調(diào)遞增區(qū)間.【變式1】（2023上·河南南陽·高三?？茧A段練習）已知函數(shù)SKIPIF1<0．(1)當SKIPIF1<0時，求曲線SKIPIF1<0在點SKIPIF1<0處的切線方程；(2)當SKIPIF1<0時，討論函數(shù)SKIPIF1<0的單調(diào)性；【變式2】（2023上·北京順義·高三楊鎮(zhèn)第一中學校考階段練習）已知函數(shù)SKIPIF1<0.(1)當SKIPIF1<0時，求函數(shù)SKIPIF1<0的最小值；(2)當SKIPIF1<0時，討論SKIPIF1<0的單調(diào)性.【變式2】（2023·全國·高三專題練習）已知SKIPIF1<0，求SKIPIF1<0的單調(diào)遞減區(qū)間.題型09含參問題討論單調(diào)性(導函數(shù)有效部分是二次型且不可因式分解)【典例1】（2023·全國·高三專題練習）設(shè)函數(shù)SKIPIF1<0，其中SKIPIF1<0為常數(shù)，討論函數(shù)SKIPIF1<0的單調(diào)性．【典例2】（2023·全國·高二專題練習）已知函數(shù)SKIPIF1<0.討論SKIPIF1<0的單調(diào)性．【變式1】（2022·甘肅臨夏·統(tǒng)考一模）已知函數(shù)SKIPIF1<0．(1)討論SKIPIF1<0的單調(diào)性；【變式2】（2023·全國·高二專題練習）已知函數(shù)SKIPIF1<0．討論當SKIPIF1<0時，SKIPIF1<0單調(diào)性．A夯實基礎(chǔ)B能力提升C綜合素養(yǎng)A夯實基礎(chǔ)一、單選題1．（2023上·高二課時練習）函數(shù)SKIPIF1<0的單調(diào)遞減區(qū)間是（

）A．SKIPIF1<0,SKIPIF1<0 B．SKIPIF1<0,SKIPIF1<0 C．SKIPIF1<0,SKIPIF1<0 D．SKIPIF1<0,SKIPIF1<02．（2023上·黑龍江雙鴨山·高二雙鴨山一中校考期末）函數(shù)SKIPIF1<0?的單調(diào)遞增區(qū)間是（

）A．SKIPIF1<0B．SKIPIF1<0?和SKIPIF1<0?C．SKIPIF1<0?D．SKIPIF1<0?3．（2021上·陜西漢中·高三校聯(lián)考階段練習）已知函數(shù)SKIPIF1<0的導函數(shù)SKIPIF1<0的圖象如圖所示，那么函數(shù)SKIPIF1<0SKIPIF1<0（

）A．在SKIPIF1<0上單調(diào)遞增 B．在SKIPIF1<0上單調(diào)遞減C．在SKIPIF1<0上單調(diào)遞增 D．在SKIPIF1<0上單調(diào)遞減4．（2022上·河南安陽·高三統(tǒng)考期中）已知函數(shù)SKIPIF1<0若SKIPIF1<0在SKIPIF1<0上單調(diào)遞減，則實數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<05．（2022上·四川成都·高三統(tǒng)考階段練習）若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞增，則實數(shù)k的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．（2022下·四川成都·高二四川省成都市新都一中校聯(lián)考期中）已知函數(shù)SKIPIF1<0在SKIPIF1<0上為單調(diào)遞增函數(shù)，則實數(shù)SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．（2023下·重慶江北·高二重慶十八中?？计谥校┤艉瘮?shù)SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)存在單調(diào)遞減區(qū)間，則實數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<08．（2023下·河北唐山·高二校聯(lián)考期中）已知函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，則實數(shù)a的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題9．（2021上·廣東梅州·高二統(tǒng)考期末）設(shè)SKIPIF1<0，SKIPIF1<0都是單調(diào)函數(shù)，其導函數(shù)分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，下列命題中，正確的是（

）A．若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0單調(diào)遞增；B．若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0單調(diào)遞增；C．SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0單調(diào)遞減；D．若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0單調(diào)遞減；三、填空題10．（2023·遼寧鞍山·鞍山一中?？级＃┤艉瘮?shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞減，則實數(shù)a的取值范圍是．11．（2023上·江蘇淮安·高三江蘇省清浦中學校聯(lián)考階段練習）已知函數(shù)SKIPIF1<0，若SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞增，則實數(shù)SKIPIF1<0的取值范圍是.12．（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上不單調(diào)，則實數(shù)a的取值范圍是.四、解答題13．（2022下·重慶璧山·高二重慶市璧山來鳳中學校?？茧A段練習）已知函數(shù)SKIPIF1<0．(1)當SKIPIF1<0時，求曲線SKIPIF1<0在SKIPIF1<0處的切線方程；(2)討論函數(shù)SKIPIF1<0的單調(diào)性；14．（2023上·湖北·高三隨州市曾都區(qū)第一中學校聯(lián)考期中）已知函數(shù)SKIPIF1<0.(1)若曲線SKIPIF1<0在點SKIPIF1<0處的切線與直線SKIPIF1<0平行，求出這條切線的方程；(2)討論函數(shù)SKIPIF1<0的單調(diào)性.B能力提升1．（2022上·山東日照·高三山東省日照實驗高級中學校考階段練習）若對任意的SKIPIF1<0，且當SKIPIF1<0時，都有SKIPIF1<0