新高考數(shù)學二輪復(fù)習 題型歸納演練專題3-9 利用導(dǎo)函數(shù)研究極值點偏移問題原卷版

專題3-9利用導(dǎo)函數(shù)研究極值點偏移問題目錄TOC\o"1-1"\h\u專題3-9利用導(dǎo)函數(shù)研究極值點偏移問題 1 1題型一：對稱化構(gòu)造 1題型二：比值代換法 13題型三：對數(shù)均值不等式法 22 29題型一：對稱化構(gòu)造【典例分析】例題1．（2022·江蘇南通·高三期中）已知SKIPIF1<0，其極小值為-4.(1)求SKIPIF1<0的值；(2)若關(guān)于SKIPIF1<0的方程SKIPIF1<0在SKIPIF1<0上有兩個不相等的實數(shù)根SKIPIF1<0，SKIPIF1<0，求證：SKIPIF1<0.例題2．（2022·北京市房山區(qū)良鄉(xiāng)中學高三期中）已知函數(shù)SKIPIF1<0(1)求函數(shù)SKIPIF1<0單調(diào)區(qū)間；(2)設(shè)函數(shù)SKIPIF1<0，若SKIPIF1<0是函數(shù)SKIPIF1<0的兩個零點，①求SKIPIF1<0的取值范圍；②求證：SKIPIF1<0．【提分秘籍】主要用來解決與兩個極值點之和，積相關(guān)的不等式的證明問題．其解題要點如下：(1)定函數(shù)(極值點為SKIPIF1<0)，即利用導(dǎo)函數(shù)符號的變化判斷函數(shù)的單調(diào)性，進而確定函數(shù)的極值點SKIPIF1<0．(2)構(gòu)造函數(shù)，即對結(jié)論SKIPIF1<0型，構(gòu)造函數(shù)SKIPIF1<0或SKIPIF1<0；(3)對結(jié)論SKIPIF1<0型，構(gòu)造函數(shù)SKIPIF1<0，通過研究SKIPIF1<0的單調(diào)性獲得不等式．(4)判斷單調(diào)性，即利用導(dǎo)數(shù)討論SKIPIF1<0的單調(diào)性．(5)比較大小，即判斷函數(shù)SKIPIF1<0在某段區(qū)間上的正負，并得出SKIPIF1<0與SKIPIF1<0的大小關(guān)系．(6)轉(zhuǎn)化，即利用函數(shù)SKIPIF1<0的單調(diào)性，將SKIPIF1<0與SKIPIF1<0的大小關(guān)系轉(zhuǎn)化為SKIPIF1<0與SKIPIF1<0之間的關(guān)系，進而得到所證或所求．【變式演練】1．（2022·福建·廈門外國語學校高二期末）已知函數(shù)SKIPIF1<0(1)若對任意的SKIPIF1<0，都有SKIPIF1<0恒成立，求實數(shù)SKIPIF1<0的取值范圍；(2)設(shè)SKIPIF1<0是兩個不相等的實數(shù)，且SKIPIF1<0．求證：SKIPIF1<02．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0.(1)求SKIPIF1<0的極值.(2)若SKIPIF1<0，SKIPIF1<0，證明：SKIPIF1<0.3．（2022·河北·開灤第二中學高二期末）設(shè)函數(shù)SKIPIF1<0．（1）當SKIPIF1<0有極值時，若存在SKIPIF1<0，使得SKIPIF1<0成立，求實數(shù)SKIPIF1<0的取值范圍；（2）當SKIPIF1<0時，若在SKIPIF1<0定義域內(nèi)存在兩實數(shù)SKIPIF1<0滿足SKIPIF1<0且SKIPIF1<0，證明：SKIPIF1<0．4．（2022·全國·高三階段練習（文））已知函數(shù)SKIPIF1<0（SKIPIF1<0且SKIPIF1<0）.(1)若函數(shù)SKIPIF1<0的最小值為2，求SKIPIF1<0的值；(2)在（1）的條件下，若關(guān)于SKIPIF1<0的方程SKIPIF1<0有兩個不同的實數(shù)根SKIPIF1<0，且SKIPIF1<0，求證：SKIPIF1<0.題型二：比值代換法【典例分析】例題1．（2022·全國·高二期末）已知函數(shù)SKIPIF1<0.(1)討論SKIPIF1<0的單調(diào)性.(2)若函數(shù)SKIPIF1<0有兩個零點SKIPIF1<0，且SKIPIF1<0，證明：SKIPIF1<0.例題2．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0.(1)設(shè)函數(shù)SKIPIF1<0，且SKIPIF1<0恒成立，求實數(shù)SKIPIF1<0的取值范圍；(2)求證：SKIPIF1<0；(3)設(shè)函數(shù)SKIPIF1<0的兩個零點SKIPIF1<0、SKIPIF1<0，求證：SKIPIF1<0.【提分秘籍】比值換元的目的也是消參、減元，就是根據(jù)已知條件首先建立極值點之間的關(guān)系，然后利用兩個極值點的比值作為變量，從而實現(xiàn)消參、減元的目的．設(shè)法用比值(一般用SKIPIF1<0表示)表示兩個極值點，即SKIPIF1<0，化為單變量的函數(shù)不等式，繼而將所求解問題轉(zhuǎn)化為關(guān)于SKIPIF1<0的函數(shù)問題求解．【變式演練】1．（2022·四川成都·高三期中（文））已知函數(shù)SKIPIF1<0有兩個零點SKIPIF1<0，SKIPIF1<0.（1）求a的取值范圍；（2）求證：SKIPIF1<0.2．（2022·全國·高三專題練習）設(shè)函數(shù)SKIPIF1<0.（1）若SKIPIF1<0，求SKIPIF1<0的單調(diào)區(qū)間；（2）若SKIPIF1<0存在三個極值點SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，求k的取值范圍，并證明：SKIPIF1<0.3．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0，且SKIPIF1<0是函數(shù)SKIPIF1<0的導(dǎo)函數(shù)，(1)求函數(shù)SKIPIF1<0的極值；(2)當SKIPIF1<0時，若方程SKIPIF1<0有兩個不等實根SKIPIF1<0．（?。┳C明：SKIPIF1<0；（ⅱ）證明：SKIPIF1<0．題型三：對數(shù)均值不等式法【典例分析】例題1．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0SKIPIF1<0（SKIPIF1<0為SKIPIF1<0的導(dǎo)函數(shù)）.(1)討論SKIPIF1<0單調(diào)性；(2)設(shè)SKIPIF1<0是SKIPIF1<0的兩個極值點，證明：SKIPIF1<0.例題2．（2022·黑龍江·牡丹江市第二高級中學高三階段練習）已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0在SKIPIF1<0上單調(diào)遞減，求實數(shù)SKIPIF1<0的取值范圍．(2)若SKIPIF1<0是方程SKIPIF1<0的兩個不相等的實數(shù)根，證明：SKIPIF1<0．【提分秘籍】兩個正數(shù)SKIPIF1<0和SKIPIF1<0的對數(shù)平均定義：SKIPIF1<0對數(shù)平均與算術(shù)平均、幾何平均的大小關(guān)系：SKIPIF1<0（此式記為對數(shù)平均不等式）取等條件：當且僅當SKIPIF1<0時，等號成立．【變式演練】1．（2022·全國·高三專題練習）已知函數(shù)f(x)＝lnx﹣ax，a為常數(shù)．（1）若函數(shù)f(x)在x＝1處的切線與x軸平行，求a的值；（2）當a＝1時，試比較f（m）與f（SKIPIF1<0）的大小；（3）若函數(shù)f(x)有兩個零點x1、x2，試證明x1x2＞e2．2．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0存在兩個零點SKIPIF1<0，SKIPIF1<0.（1）求SKIPIF1<0的取值范圍；（2）證明：SKIPIF1<0.一、單選題1．（2022·吉林長春·模擬預(yù)測）已知a，b滿足SKIPIF1<0，SKIPIF1<0，其中e是自然對數(shù)的底數(shù)，則ab的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0，對于正實數(shù)a，若關(guān)于t的方程SKIPIF1<0恰有三個不同的正實數(shù)根，則a的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2021·河南·鄭州外國語中學高三階段練習（理））關(guān)于函數(shù)SKIPIF1<0，下列說法錯誤的是（

）A．SKIPIF1<0是SKIPIF1<0的極小值點B．函數(shù)SKIPIF1<0有且只有SKIPIF1<0個零點C．存在正實數(shù)SKIPIF1<0，使得SKIPIF1<0恒成立D．對任意兩個正實數(shù)SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<04．（2021·江西·鷹潭一中高三階段練習（文））關(guān)于函數(shù)SKIPIF1<0，下列說法正確的是（

）A．SKIPIF1<0是SKIPIF1<0的極大值點B．函數(shù)SKIPIF1<0有2個零點C．存在正整數(shù)k，使得SKIPIF1<0恒成立D．對任意兩個正實數(shù)SKIPIF1<0，且SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0二、多選題5．（2022·黑龍江·哈爾濱市第六中學校高三期中）已知函數(shù)SKIPIF1<0則下列結(jié)論正確的有（

）A．當SKIPIF1<0時，SKIPIF1<0是SKIPIF1<0的極值點B．當SKIPIF1<0時，SKIPIF1<0恒成立C．當SKIPIF1<0時，SKIPIF1<0有2個零點D．若SKIPIF1<0是關(guān)于x的方程SKIPIF1<0的2個不等實數(shù)根，則SKIPIF1<06．（2022·黑龍江·哈爾濱三中模擬預(yù)測）已知函數(shù)SKIPIF1<0，則下列說法正確的是（

）A．若SKIPIF1<0恒成立，則SKIPIF1<0B．當SKIPIF1<0時，SKIPIF1<0的零點只有SKIPIF1<0個C．若函數(shù)SKIPIF1<0有兩個不同的零點SKIPIF1<0，則SKIPIF1<0D．當SKIPIF1<0時，若不等式SKIPIF1<0恒成立，則正數(shù)SKIPIF1<0的取值范圍是SKIPIF1<07．（2022·全國·高二專題練習）已知函數(shù)SKIPIF1<0，則（

）A．SKIPIF1<0B．若SKIPIF1<0有兩個不相等的實根SKIPIF1<0、SKIPIF1<0，則SKIPIF1<0C．SKIPIF1<0D．若SKIPIF1<0，x，y均為正數(shù)，則SKIPIF1<0三、解答題8．（2022·湖南·長沙市同升湖高級中學有限公司高三階段練習）已知函數(shù)SKIPIF1<0.(1)證明：SKIPIF1<0.(2)若函數(shù)SKIPIF1<0，若存在SKIPIF1<0使SKIPIF1<0，證明：SKIPIF1<0.9．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0.(1)求SKIPIF1<0的單調(diào)區(qū)間(2)若SKIPIF1<0的極值點為SKIPIF1<0，且SKIPIF1<0，證明：SKIPIF1<0.10．（2022·江蘇常州·高三期中）已知函數(shù)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0．(1)若SKIPIF1<0在x＝0處的切線與SKIPIF1<0在x＝1處的切線相同，求實數(shù)a的值；(2)令SKIPIF1<0