新高考數(shù)學二輪復習 題型歸納演練專題3-6 利用導函數(shù)研究方程的根（函數(shù)的零點）原卷版

專題3-6利用導函數(shù)研究方程的根（函數(shù)的零點）目錄TOC\o"1-1"\h\u 1題型一：判斷（證明）函數(shù)零點個數(shù) 1題型二：利用函數(shù)極值（最值）研究函數(shù)的零點 9題型三：已知函數(shù)的零點個數(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（等價為兩個函數(shù)圖象交點）研究函數(shù)的零點（方程的根） 22題型五：以函數(shù)零點為背景的含雙參不等式的證明 32題型六：導數(shù)解決函數(shù)隱零點問題 44 50題型一：判斷（證明）函數(shù)零點個數(shù)【典例分析】例題1．（2022·河南·駐馬店開發(fā)區(qū)高級中學高三階段練習（文））已知函數(shù)SKIPIF1<0圖象的對稱中心為SKIPIF1<0，則SKIPIF1<0的零點個數(shù)為（

）A．2 B．1 C．4 D．3例題2．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0，討論函數(shù)SKIPIF1<0的零點的個數(shù).例題3．（2022·安徽·安慶一中高三階段練習（理））已知函數(shù)SKIPIF1<0.(1)若SKIPIF1<0的圖象在點SKIPIF1<0處的切線斜率為SKIPIF1<0，求SKIPIF1<0的值；(2)當SKIPIF1<0時，判斷SKIPIF1<0在SKIPIF1<0內(nèi)有幾個零點，并證明.【提分秘籍】1.利用導數(shù)研究高次式、分式、指數(shù)式、對數(shù)式、三角式及絕對值式等函數(shù)零點的個數(shù)(或方程根的個數(shù))問題的一般思路:(1)可轉(zhuǎn)化為用導數(shù)研究其函數(shù)的圖象與SKIPIF1<0軸(或直線SKIPIF1<0)在該區(qū)間上的交點問題;(2)利用導數(shù)研究該函數(shù)在該區(qū)間上的單調(diào)性、極值(最值)、端點值等性質(zhì),進而畫出其圖象;(3)結(jié)合圖象求解.2.證明復雜方程在某區(qū)間上有且僅有一解的步驟:第一步,利用導數(shù)證明該函數(shù)在該區(qū)間上單調(diào)性，第二步,證明端點的導數(shù)值異號.【變式演練】1．（2022·湖南·高三階段練習）已知函數(shù)SKIPIF1<0，則函數(shù)SKIPIF1<0的零點個數(shù)為_________．2．（2022·河南南陽·高二階段練習（理））已知函數(shù)SKIPIF1<0，SKIPIF1<0．(1)求證：函數(shù)SKIPIF1<0有唯一的零點，并求出此零點；(2)求曲線SKIPIF1<0過點SKIPIF1<0的切線方程．3．（2022·全國·高二專題練習）已知函數(shù)SKIPIF1<0.(1)求曲線y=f（x）在點（0，f（0））處的切線方程；(2)判斷函數(shù)f（x）的零點的個數(shù)，并說明理由.4．（2022·全國·成都七中高三開學考試（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?的零點個數(shù).5．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0，其中SKIPIF1<0.(1)求SKIPIF1<0的極值點個數(shù)；(2)求函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)的零點個數(shù).題型二：利用函數(shù)極值（最值）研究函數(shù)的零點【典例分析】例題1．（2022·四川·雅安中學高二階段練習（文））已知函數(shù)SKIPIF1<0在SKIPIF1<0時取得極值，且在點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有三個零點，求實數(shù)SKIPIF1<0的取值范圍.例題2．（2022·寧夏·銀川一中模擬預測（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上有且只有一個零點，求SKIPIF1<0在SKIPIF1<0上的最大值與最小值的和．【提分秘籍】借助導數(shù)研究函數(shù)的單調(diào)性與極值后,通過極值（最值）的正負,函數(shù)的單調(diào)性判斷函數(shù)圖象的走勢,從而判斷零點的個數(shù).【變式演練】1．（2022·重慶八中高二階段練習）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有三個零點，求實數(shù)a的取值范圍．2．（2022·廣東·高二階段練習）已知函數(shù)SKIPIF1<0在SKIPIF1<0與SKIPIF1<0處都取得極值.(1)求實數(shù)a，b的值；(2)若函數(shù)SKIPIF1<0有三個不同的零點，求c的范圍.3．（2022·全國·模擬預測（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個零點，求SKIPIF1<0的取值范圍．題型三：已知函數(shù)的零點個數(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（文））已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的奇函數(shù)，且當SKIPIF1<0時，SKIPIF1<0，若關(guān)于SKIPIF1<0的函數(shù)SKIPIF1<0恰有4個零點，則實數(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·武功縣普集高級中學模擬預測（理））已知關(guān)于SKIPIF1<0的方程SKIPIF1<0有4個不等實數(shù)根，則SKIPIF1<0的取值范圍是______．例題3．（2022·湖南·長郡中學高二階段練習）已知函數(shù)SKIPIF1<0.(1)當SKIPIF1<0時，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上有兩個零點，求實數(shù)SKIPIF1<0的取值范圍.【提分秘籍】轉(zhuǎn)化為用導數(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有兩個零點，則實數(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）已知函數(shù)SKIPIF1<0，若f（x）在（0，+∞）內(nèi)有零點，則a的取值范圍為___________.3．（2022·河南·高三階段練習（文））若函數(shù)SKIPIF1<0有且只有一個零點，則實數(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恰有兩個零點，求實數(shù)m的取值范圍.題型四：利用數(shù)形結(jié)合法（等價為兩個函數(shù)圖象交點）研究函數(shù)的零點（方程的根）【典例分析】例題1．（2022·河北石家莊·高二階段練習）已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0，求曲線SKIPIF1<0在點SKIPIF1<0處的切線方程；(2)若方程SKIPIF1<0有三個不同的根，求SKIPIF1<0的取值范圍．例題2．（2022·重慶市永川北山中學校模擬預測）已知函數(shù)SKIPIF1<0，SKIPIF1<0(1)當SKIPIF1<0時，求SKIPIF1<0的極值；(2)若SKIPIF1<0，函數(shù)SKIPIF1<0與SKIPIF1<0軸有兩個交點，求SKIPIF1<0的取值范圍.例題3．（2022·山東·寧陽縣第四中學高二階段練習）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：坐標系中要標出關(guān)鍵點）；(3)求出方程SKIPIF1<0的解的個數(shù).【提分秘籍】轉(zhuǎn)化為用導數(shù)研究其函數(shù)的圖象與SKIPIF1<0軸(或直線SKIPIF1<0)在該區(qū)間上的交點問題;【變式演練】1．（2022·全國·高三專題練習）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上有兩個零點，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SKIPIF1<0處的切線方程；(2)若方程SKIPIF1<0有兩個根，求a的取值范圍．3．（2022·廣東·珠海市第二中學高二期中）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有三個不同的解，求a的取值范圍.4．（2022·全國·高三專題練習）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僅有一個公共點，求實數(shù)SKIPIF1<0的取值范圍.題型五：以函數(shù)零點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有兩個零點SKIPIF1<0，SKIPIF1<0，求SKIPIF1<0的取值范圍．例題2．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0有兩個不同的零點(其中SKIPIF1<0為自然對數(shù)的底數(shù)).（1）當SKIPIF1<0時，求證：SKIPIF1<0；（2）求實數(shù)SKIPIF1<0的取值范圍；（3）若函數(shù)SKIPIF1<0的兩個零點為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有兩個零點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·高三專題練習（理））已知SKIPIF1<0，設(shè)函數(shù)SKIPIF1<0．(1)當SKIPIF1<0時，若函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，求實數(shù)SKIPIF1<0的取值范圍；(2)若對任意實數(shù)SKIPIF1<0，函數(shù)SKIPIF1<0均有零點，求實數(shù)SKIPIF1<0的最大值；(3)若函數(shù)SKIPIF1<0有兩個零點SKIPIF1<0，證明：SKIPIF1<0．2．（2022·全國·高二課時練習）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存在兩個零點（記為SKIPIF1<0），且SKIPIF1<0．3．（2022·全國·高三專題練習）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的導函數(shù)，若函數(shù)SKIPIF1<0有兩個不同的零點SKIPIF1<0，求證：SKIPIF1<0．4．（2022·全國·高三專題練習）已知函數(shù)SKIPIF1<0（a為常數(shù)）．且SKIPIF1<0有兩個不同的極值點SKIPIF1<0（1）求實數(shù)a的取值范圍；（2）求證：SKIPIF1<0題型六：導數(shù)解決函數(shù)隱零點問題【典例分析】例題1．（2022·全國·高二單元測試）設(shè)SKIPIF1<0，SKIPIF1<0.（1）求SKIPIF1<0的單調(diào)區(qū)間；（2）討論SKIPIF1<0零點的個數(shù)；（3）當SKIPIF1<0時，設(shè)SKIPIF1<0恒成立，求實數(shù)SKIPIF1<0的取值范圍.例題2．（2022·遼寧·東北育才雙語學校一模）已知函數(shù)SKIPIF1<0.（1）當SKIPIF1<0時，求SKIPIF1<0的圖象在點SKIPIF1<0處的切線方程；（2）當SKIPIF1<0時，判斷SKIPIF1<0的零點個數(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)隱零點在很多時候無法直接求出來，基本解決思路是：虛設(shè)零點，整體代換，數(shù)值估算，等價轉(zhuǎn)化，分離參數(shù)，反客為主?！咀兪窖菥殹?．（2022·全國·高二專題練習）已知函數(shù)SKIPIF1<0．（Ⅰ）求函數(shù)SKIPIF1<0的零點及單調(diào)區(qū)間；（Ⅱ）求證：曲線SKIPIF1<0存在斜率為SKIPIF1<0的切線，且切點的縱坐標SKIPIF1<0．2．（2022·北京·北師大二附中高三階段練習）已知函數(shù)SKIPIF1<0，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，求SKIPIF1<0在區(qū)間SKIPIF1<0上的最大值和最小值；（3）當SKIPIF1<0時，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·上海市楊浦高級中學高三開學考試）已知點P是曲線SKIPIF1<0上任意一點，記直線OP（O為坐標系原點）的斜率為k，則使得SKIPIF1<0的點P的個數(shù)為（

）.A．0 B．僅有1個 C．僅有2個 D．至少有3個2．（2022·全國·高三專題練習）設(shè)SKIPIF1<0，若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上有三個零點，則實數(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有兩個零點SKIPIF1<0，對于下列結(jié)論：①SKIPIF1<0；②SKIPIF1<0；則（

）A．①②均對 B．①②均錯 C．①對②錯 D．①錯②對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個不同的交點，則實數(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個零點，則實數(shù)m的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<06．（2022·云南·昆明市第三中學高三階段練習）過點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）直線SKIPIF1<0與兩條曲線SKIPIF1<0和SKIPIF1<0共有三個不同的交點，并且從左到右三個交點的橫坐標依次是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·晉城一中教育集團南嶺愛物學校高三階段練習）已知當SKIPIF1<0時，函數(shù)SKIPIF1<0的圖像與函數(shù)SKIPIF1<0的圖像有且只有兩個交點，則實數(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·高三專題練習）已知函數(shù)SKIPIF1<0有唯一零點，則實數(shù)SKIPIF1<0的值可以是（

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

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