新高考數(shù)學(xué)二輪復(fù)習(xí) 題型歸納演練專題3-6 利用導(dǎo)函數(shù)研究方程的根（函數(shù)的零點(diǎn)）原卷版

專題3-6利用導(dǎo)函數(shù)研究方程的根（函數(shù)的零點(diǎn)）目錄TOC\o"1-1"\h\u 1題型一：判斷（證明）函數(shù)零點(diǎn)個(gè)數(shù) 1題型二：利用函數(shù)極值（最值）研究函數(shù)的零點(diǎn) 9題型三：已知函數(shù)的零點(diǎn)個(gè)數(shù)求參數(shù)的取值范圍(或值) 15題型四：利用數(shù)形結(jié)合法（等價(jià)為兩個(gè)函數(shù)圖象交點(diǎn)）研究函數(shù)的零點(diǎn)（方程的根） 22題型五：以函數(shù)零點(diǎn)為背景的含雙參不等式的證明 32題型六：導(dǎo)數(shù)解決函數(shù)隱零點(diǎn)問題 44 50題型一：判斷（證明）函數(shù)零點(diǎn)個(gè)數(shù)【典例分析】例題1．（2022·河南·駐馬店開發(fā)區(qū)高級(jí)中學(xué)高三階段練習(xí)（文））已知函數(shù)SKIPIF1<0圖象的對(duì)稱中心為SKIPIF1<0，則SKIPIF1<0的零點(diǎn)個(gè)數(shù)為（

）A．2 B．1 C．4 D．3例題2．（2022·全國(guó)·高三專題練習(xí)）已知函數(shù)SKIPIF1<0，討論函數(shù)SKIPIF1<0的零點(diǎn)的個(gè)數(shù).例題3．（2022·安徽·安慶一中高三階段練習(xí)（理））已知函數(shù)SKIPIF1<0.(1)若SKIPIF1<0的圖象在點(diǎn)SKIPIF1<0處的切線斜率為SKIPIF1<0，求SKIPIF1<0的值；(2)當(dāng)SKIPIF1<0時(shí)，判斷SKIPIF1<0在SKIPIF1<0內(nèi)有幾個(gè)零點(diǎn)，并證明.【提分秘籍】1.利用導(dǎo)數(shù)研究高次式、分式、指數(shù)式、對(duì)數(shù)式、三角式及絕對(duì)值式等函數(shù)零點(diǎn)的個(gè)數(shù)(或方程根的個(gè)數(shù))問題的一般思路:(1)可轉(zhuǎn)化為用導(dǎo)數(shù)研究其函數(shù)的圖象與SKIPIF1<0軸(或直線SKIPIF1<0)在該區(qū)間上的交點(diǎn)問題;(2)利用導(dǎo)數(shù)研究該函數(shù)在該區(qū)間上的單調(diào)性、極值(最值)、端點(diǎn)值等性質(zhì),進(jìn)而畫出其圖象;(3)結(jié)合圖象求解.2.證明復(fù)雜方程在某區(qū)間上有且僅有一解的步驟:第一步,利用導(dǎo)數(shù)證明該函數(shù)在該區(qū)間上單調(diào)性，第二步,證明端點(diǎn)的導(dǎo)數(shù)值異號(hào).【變式演練】1．（2022·湖南·高三階段練習(xí)）已知函數(shù)SKIPIF1<0，則函數(shù)SKIPIF1<0的零點(diǎn)個(gè)數(shù)為_________．2．（2022·河南南陽·高二階段練習(xí)（理））已知函數(shù)SKIPIF1<0，SKIPIF1<0．(1)求證：函數(shù)SKIPIF1<0有唯一的零點(diǎn)，并求出此零點(diǎn)；(2)求曲線SKIPIF1<0過點(diǎn)SKIPIF1<0的切線方程．3．（2022·全國(guó)·高二專題練習(xí)）已知函數(shù)SKIPIF1<0.(1)求曲線y=f（x）在點(diǎn)（0，f（0））處的切線方程；(2)判斷函數(shù)f（x）的零點(diǎn)的個(gè)數(shù)，并說明理由.4．（2022·全國(guó)·成都七中高三開學(xué)考試（文））設(shè)函數(shù)SKIPIF1<0?為常數(shù)).(1)討論SKIPIF1<0?的單調(diào)性；(2)討論函數(shù)SKIPIF1<0?的零點(diǎn)個(gè)數(shù).5．（2022·全國(guó)·高三專題練習(xí)）已知函數(shù)SKIPIF1<0，其中SKIPIF1<0.(1)求SKIPIF1<0的極值點(diǎn)個(gè)數(shù)；(2)求函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)的零點(diǎn)個(gè)數(shù).題型二：利用函數(shù)極值（最值）研究函數(shù)的零點(diǎn)【典例分析】例題1．（2022·四川·雅安中學(xué)高二階段練習(xí)（文））已知函數(shù)SKIPIF1<0在SKIPIF1<0時(shí)取得極值，且在點(diǎn)SKIPIF1<0處的切線的斜率為SKIPIF1<0.(1)求SKIPIF1<0的解析式；(2)若函數(shù)SKIPIF1<0有三個(gè)零點(diǎn)，求實(shí)數(shù)SKIPIF1<0的取值范圍.例題2．（2022·寧夏·銀川一中模擬預(yù)測(cè)（文））已知函數(shù)SKIPIF1<0．(1)討論SKIPIF1<0的單調(diào)性；(2)若SKIPIF1<0在SKIPIF1<0上有且只有一個(gè)零點(diǎn)，求SKIPIF1<0在SKIPIF1<0上的最大值與最小值的和．【提分秘籍】借助導(dǎo)數(shù)研究函數(shù)的單調(diào)性與極值后,通過極值（最值）的正負(fù),函數(shù)的單調(diào)性判斷函數(shù)圖象的走勢(shì),從而判斷零點(diǎn)的個(gè)數(shù).【變式演練】1．（2022·重慶八中高二階段練習(xí)）已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0，求函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的最大值；(2)若函數(shù)SKIPIF1<0有三個(gè)零點(diǎn)，求實(shí)數(shù)a的取值范圍．2．（2022·廣東·高二階段練習(xí)）已知函數(shù)SKIPIF1<0在SKIPIF1<0與SKIPIF1<0處都取得極值.(1)求實(shí)數(shù)a，b的值；(2)若函數(shù)SKIPIF1<0有三個(gè)不同的零點(diǎn)，求c的范圍.3．（2022·全國(guó)·模擬預(yù)測(cè)（文））設(shè)函數(shù)SKIPIF1<0，其中SKIPIF1<0，SKIPIF1<0為常數(shù)．(1)討論SKIPIF1<0的單調(diào)性；(2)若函數(shù)SKIPIF1<0有且僅有3個(gè)零點(diǎn)，求SKIPIF1<0的取值范圍．題型三：已知函數(shù)的零點(diǎn)個(gè)數(shù)求參數(shù)的取值范圍(或值)【典例分析】例題1．（2022·貴州·貴陽一中高三階段練習(xí)（文））已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，且當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，若關(guān)于SKIPIF1<0的函數(shù)SKIPIF1<0恰有4個(gè)零點(diǎn)，則實(shí)數(shù)SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例題2．（2022·全國(guó)·武功縣普集高級(jí)中學(xué)模擬預(yù)測(cè)（理））已知關(guān)于SKIPIF1<0的方程SKIPIF1<0有4個(gè)不等實(shí)數(shù)根，則SKIPIF1<0的取值范圍是______．例題3．（2022·湖南·長(zhǎng)郡中學(xué)高二階段練習(xí)）已知函數(shù)SKIPIF1<0.(1)當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0的圖像在SKIPIF1<0處的切線方程；(2)若函數(shù)SKIPIF1<0在SKIPIF1<0上有兩個(gè)零點(diǎn)，求實(shí)數(shù)SKIPIF1<0的取值范圍.【提分秘籍】轉(zhuǎn)化為用導(dǎo)數(shù)研究其函數(shù)的圖象與SKIPIF1<0軸(或直線SKIPIF1<0)在該區(qū)間上的交點(diǎn)問題;【變式演練】1．（2022·北京通州·高三期中）已知函數(shù)SKIPIF1<0設(shè)SKIPIF1<0，若函數(shù)SKIPIF1<0有兩個(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<02．（2022·廣東·順德一中高三階段練習(xí)）已知函數(shù)SKIPIF1<0，若f（x）在（0，+∞）內(nèi)有零點(diǎn)，則a的取值范圍為___________.3．（2022·河南·高三階段練習(xí)（文））若函數(shù)SKIPIF1<0有且只有一個(gè)零點(diǎn)，則實(shí)數(shù)SKIPIF1<0的取值范圍是___________.4．（2022·天津·高三期中）已知函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線斜率為4，且在SKIPIF1<0處取得極值.(1)求函數(shù)SKIPIF1<0的單調(diào)區(qū)間；(2)若函數(shù)SKIPIF1<0恰有兩個(gè)零點(diǎn)，求實(shí)數(shù)m的取值范圍.題型四：利用數(shù)形結(jié)合法（等價(jià)為兩個(gè)函數(shù)圖象交點(diǎn)）研究函數(shù)的零點(diǎn)（方程的根）【典例分析】例題1．（2022·河北石家莊·高二階段練習(xí)）已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0，求曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程；(2)若方程SKIPIF1<0有三個(gè)不同的根，求SKIPIF1<0的取值范圍．例題2．（2022·重慶市永川北山中學(xué)校模擬預(yù)測(cè)）已知函數(shù)SKIPIF1<0，SKIPIF1<0(1)當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0的極值；(2)若SKIPIF1<0，函數(shù)SKIPIF1<0與SKIPIF1<0軸有兩個(gè)交點(diǎn)，求SKIPIF1<0的取值范圍.例題3．（2022·山東·寧陽縣第四中學(xué)高二階段練習(xí)）給定函數(shù)SKIPIF1<0.(1)判斷函數(shù)SKIPIF1<0的單調(diào)性，并求出SKIPIF1<0的極值；(2)畫出函數(shù)SKIPIF1<0的大致圖象，無須說明理由（要求：坐標(biāo)系中要標(biāo)出關(guān)鍵點(diǎn)）；(3)求出方程SKIPIF1<0的解的個(gè)數(shù).【提分秘籍】轉(zhuǎn)化為用導(dǎo)數(shù)研究其函數(shù)的圖象與SKIPIF1<0軸(或直線SKIPIF1<0)在該區(qū)間上的交點(diǎn)問題;【變式演練】1．（2022·全國(guó)·高三專題練習(xí)）已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0，求曲線SKIPIF1<0在SKIPIF1<0處的切線方程；(2)若函數(shù)SKIPIF1<0在SKIPIF1<0上有兩個(gè)零點(diǎn)，求SKIPIF1<0的取值范圍．2．（2022·遼寧·高二期中）已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0，求曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程；(2)若方程SKIPIF1<0有兩個(gè)根，求a的取值范圍．3．（2022·廣東·珠海市第二中學(xué)高二期中）已知函數(shù)SKIPIF1<0(1)討論SKIPIF1<0的單調(diào)性；(2)設(shè)SKIPIF1<0，若方程SKIPIF1<0有三個(gè)不同的解，求a的取值范圍.4．（2022·全國(guó)·高三專題練習(xí)）已知函數(shù)SKIPIF1<0.(1)求函數(shù)SKIPIF1<0的單調(diào)區(qū)間和極值；(2)若函數(shù)SKIPIF1<0的圖象與直線SKIPIF1<0僅有一個(gè)公共點(diǎn)，求實(shí)數(shù)SKIPIF1<0的取值范圍.題型五：以函數(shù)零點(diǎn)為背景的含雙參不等式的證明【典例分析】例題1．（2022·河南·一模（文））已知函數(shù)SKIPIF1<0．（1）討論函數(shù)SKIPIF1<0的單調(diào)性；（2）若函數(shù)SKIPIF1<0有兩個(gè)零點(diǎn)SKIPIF1<0，SKIPIF1<0，求SKIPIF1<0的取值范圍．例題2．（2022·全國(guó)·高三專題練習(xí)）已知函數(shù)SKIPIF1<0有兩個(gè)不同的零點(diǎn)(其中SKIPIF1<0為自然對(duì)數(shù)的底數(shù)).（1）當(dāng)SKIPIF1<0時(shí)，求證：SKIPIF1<0；（2）求實(shí)數(shù)SKIPIF1<0的取值范圍；（3）若函數(shù)SKIPIF1<0的兩個(gè)零點(diǎn)為SKIPIF1<0，求證：SKIPIF1<0.例題3．（2022·江西鷹潭·高二期末（文））設(shè)函數(shù)SKIPIF1<0．(1)求函數(shù)SKIPIF1<0的單調(diào)區(qū)間；(2)若SKIPIF1<0有兩個(gè)零點(diǎn)SKIPIF1<0，SKIPIF1<0，求SKIPIF1<0的取值范圍，并證明：SKIPIF1<0．【提分秘籍】破解含雙變量不等式的證明的關(guān)鍵一是轉(zhuǎn)化,即由已知條件入手,尋找雙變量所滿足的關(guān)系式,并把含雙變量的不等式轉(zhuǎn)化為含單變量的不等式;二是巧構(gòu)造函數(shù),借助導(dǎo)數(shù),判斷函數(shù)的單調(diào)性,從而求其最值;三是回歸雙變量的不等式的證明,把所求的最值應(yīng)用到雙變量不等式,即可證得結(jié)果.【變式演練】1．（2022·全國(guó)·高三專題練習(xí)（理））已知SKIPIF1<0，設(shè)函數(shù)SKIPIF1<0．(1)當(dāng)SKIPIF1<0時(shí)，若函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，求實(shí)數(shù)SKIPIF1<0的取值范圍；(2)若對(duì)任意實(shí)數(shù)SKIPIF1<0，函數(shù)SKIPIF1<0均有零點(diǎn)，求實(shí)數(shù)SKIPIF1<0的最大值；(3)若函數(shù)SKIPIF1<0有兩個(gè)零點(diǎn)SKIPIF1<0，證明：SKIPIF1<0．2．（2022·全國(guó)·高二課時(shí)練習(xí)）已知函數(shù)SKIPIF1<0．(1)求函數(shù)SKIPIF1<0的最小值；(2)求證：函數(shù)SKIPIF1<0存在兩個(gè)零點(diǎn)（記為SKIPIF1<0），且SKIPIF1<0．3．（2022·全國(guó)·高三專題練習(xí)）已知函數(shù)SKIPIF1<0．（1）討論函數(shù)SKIPIF1<0的單調(diào)性;（2）若SKIPIF1<0，設(shè)SKIPIF1<0為SKIPIF1<0的導(dǎo)函數(shù)，若函數(shù)SKIPIF1<0有兩個(gè)不同的零點(diǎn)SKIPIF1<0，求證：SKIPIF1<0．4．（2022·全國(guó)·高三專題練習(xí)）已知函數(shù)SKIPIF1<0（a為常數(shù)）．且SKIPIF1<0有兩個(gè)不同的極值點(diǎn)SKIPIF1<0（1）求實(shí)數(shù)a的取值范圍；（2）求證：SKIPIF1<0題型六：導(dǎo)數(shù)解決函數(shù)隱零點(diǎn)問題【典例分析】例題1．（2022·全國(guó)·高二單元測(cè)試）設(shè)SKIPIF1<0，SKIPIF1<0.（1）求SKIPIF1<0的單調(diào)區(qū)間；（2）討論SKIPIF1<0零點(diǎn)的個(gè)數(shù)；（3）當(dāng)SKIPIF1<0時(shí)，設(shè)SKIPIF1<0恒成立，求實(shí)數(shù)SKIPIF1<0的取值范圍.例題2．（2022·遼寧·東北育才雙語學(xué)校一模）已知函數(shù)SKIPIF1<0.（1）當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0的圖象在點(diǎn)SKIPIF1<0處的切線方程；（2）當(dāng)SKIPIF1<0時(shí)，判斷SKIPIF1<0的零點(diǎn)個(gè)數(shù)并說明理由；（3）若SKIPIF1<0恒成立，求SKIPIF1<0的取值范圍.【提分秘籍】函數(shù)隱零點(diǎn)在很多時(shí)候無法直接求出來，基本解決思路是：虛設(shè)零點(diǎn)，整體代換，數(shù)值估算，等價(jià)轉(zhuǎn)化，分離參數(shù)，反客為主?！咀兪窖菥殹?．（2022·全國(guó)·高二專題練習(xí)）已知函數(shù)SKIPIF1<0．（Ⅰ）求函數(shù)SKIPIF1<0的零點(diǎn)及單調(diào)區(qū)間；（Ⅱ）求證：曲線SKIPIF1<0存在斜率為SKIPIF1<0的切線，且切點(diǎn)的縱坐標(biāo)SKIPIF1<0．2．（2022·北京·北師大二附中高三階段練習(xí)）已知函數(shù)SKIPIF1<0，SKIPIF1<0.（1）當(dāng)SKIPIF1<0時(shí)，求曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程；（2）當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0在區(qū)間SKIPIF1<0上的最大值和最小值；（3）當(dāng)SKIPIF1<0時(shí)，若方程SKIPIF1<0在區(qū)間SKIPIF1<0上有唯一解，求SKIPIF1<0的取值范圍.一、單選題1．（2022·上海市楊浦高級(jí)中學(xué)高三開學(xué)考試）已知點(diǎn)P是曲線SKIPIF1<0上任意一點(diǎn)，記直線OP（O為坐標(biāo)系原點(diǎn)）的斜率為k，則使得SKIPIF1<0的點(diǎn)P的個(gè)數(shù)為（

）.A．0 B．僅有1個(gè) C．僅有2個(gè) D．至少有3個(gè)2．（2022·全國(guó)·高三專題練習(xí)）設(shè)SKIPIF1<0，若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上有三個(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．（2022·上?！げ軛疃懈叨谀┮阎瘮?shù)SKIPIF1<0有兩個(gè)零點(diǎn)SKIPIF1<0，對(duì)于下列結(jié)論：①SKIPIF1<0；②SKIPIF1<0；則（

）A．①②均對(duì) B．①②均錯(cuò) C．①對(duì)②錯(cuò) D．①錯(cuò)②對(duì)4．（2022·山東德州·高三期中）已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0，若SKIPIF1<0的圖像與SKIPIF1<0軸有4個(gè)不同的交點(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<05．（2022·天津·高三期中）已知定義在R上的函數(shù)SKIPIF1<0，若函數(shù)SKIPIF1<0恰有2個(gè)零點(diǎn)，則實(shí)數(shù)m的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<06．（2022·云南·昆明市第三中學(xué)高三階段練習(xí)）過點(diǎn)SKIPIF1<0有SKIPIF1<0條直線與函數(shù)SKIPIF1<0的圖象相切，則SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<07．（2022·湖北·高三階段練習(xí)）直線SKIPIF1<0與兩條曲線SKIPIF1<0和SKIPIF1<0共有三個(gè)不同的交點(diǎn)，并且從左到右三個(gè)交點(diǎn)的橫坐標(biāo)依次是SKIPIF1<0、SKIPIF1<0、SKIPIF1<0，則下列關(guān)系式正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．（2022·山西·晉城一中教育集團(tuán)南嶺愛物學(xué)校高三階段練習(xí)）已知當(dāng)SKIPIF1<0時(shí)，函數(shù)SKIPIF1<0的圖像與函數(shù)SKIPIF1<0的圖像有且只有兩個(gè)交點(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<0二、多選題9．（2022·全國(guó)·高三專題練習(xí)）已知函數(shù)SKIPIF1<0有唯一零點(diǎn)，則實(shí)數(shù)SKIPIF1<0的值可以是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．0 D．110．（2022·湖南·益陽市箴言中學(xué)高二開學(xué)考試）已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)有唯一零點(diǎn)，則SKIPIF1<0的可能取值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIP