人教A版高中數(shù)學(xué)（選擇性必修二）同步講義第16講 拓展三：利用導(dǎo)數(shù)研究函數(shù)的零點(diǎn)（方程的根）（原卷版）

第07講拓展三：利用導(dǎo)數(shù)研究函數(shù)的零點(diǎn)（方程的根）一、知識(shí)點(diǎn)歸納1、函數(shù)的零點(diǎn)（1）函數(shù)零點(diǎn)的定義：對于函數(shù)SKIPIF1<0，把使SKIPIF1<0的實(shí)數(shù)SKIPIF1<0叫做函數(shù)SKIPIF1<0的零點(diǎn)．（2）三個(gè)等價(jià)關(guān)系方程SKIPIF1<0有實(shí)數(shù)根SKIPIF1<0函數(shù)SKIPIF1<0的圖象與SKIPIF1<0軸有交點(diǎn)的橫坐標(biāo)SKIPIF1<0函數(shù)SKIPIF1<0有零點(diǎn)．2、函數(shù)零點(diǎn)的判定如果函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的圖象是連續(xù)不斷的一條曲線，并且有SKIPIF1<0，那么函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)有零點(diǎn)，即存在SKIPIF1<0，使得SKIPIF1<0，這個(gè)SKIPIF1<0也就是SKIPIF1<0的根．我們把這一結(jié)論稱為函數(shù)零點(diǎn)存在性定理．注意：單調(diào)性+存在零點(diǎn)=唯一零點(diǎn)二、題型精講題型01判斷、證明或討論函數(shù)零點(diǎn)（方程的根）的個(gè)數(shù)1．（2022上·江蘇·高三校聯(lián)考階段練習(xí)）已知函數(shù)SKIPIF1<0.(1)設(shè)SKIPIF1<0，求SKIPIF1<0在區(qū)間SKIPIF1<0上的最值；(2)討論SKIPIF1<0的零點(diǎn)個(gè)數(shù).2．（2022下·山東青島·高二山東省萊西市第一中學(xué)?？茧A段練習(xí)）已知函數(shù)SKIPIF1<0，討論函數(shù)SKIPIF1<0的零點(diǎn)的個(gè)數(shù).3．（2022下·山東聊城·高二統(tǒng)考期末）已知函數(shù)SKIPIF1<0，在SKIPIF1<0處切線的斜率為-2.（1）求SKIPIF1<0的值及SKIPIF1<0的極小值；（2）討論方程SKIPIF1<0的實(shí)數(shù)解的個(gè)數(shù).4．（2023上·云南·高三校聯(lián)考階段練習(xí)）已知SKIPIF1<0．(1)當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0在SKIPIF1<0上的單調(diào)性；(2)若SKIPIF1<0，令SKIPIF1<0，討論方程SKIPIF1<0的解的個(gè)數(shù)．5．（2023·四川綿陽·統(tǒng)考模擬預(yù)測）函數(shù)SKIPIF1<0．(1)若SKIPIF1<0為奇函數(shù)，求實(shí)數(shù)SKIPIF1<0的值；(2)已知SKIPIF1<0僅有兩個(gè)零點(diǎn)，證明：函數(shù)SKIPIF1<0僅有一個(gè)零點(diǎn)．題型02利用最值（極值）研究函數(shù)零點(diǎn)（方程的根）問題1．（2023上·北京·高三北京四中?？计谥校┮阎瘮?shù)SKIPIF1<0.(1)當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0的極值；(2)當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0在SKIPIF1<0上的最小值；(3)若SKIPIF1<0在SKIPIF1<0上存在零點(diǎn)，求SKIPIF1<0的取值范圍.2．（2023上·福建福州·高三校聯(lián)考期中）已知函數(shù)SKIPIF1<0（SKIPIF1<0）.(1)求SKIPIF1<0在SKIPIF1<0上的最大值；(2)若函數(shù)SKIPIF1<0恰有三個(gè)零點(diǎn)，求a的取值范圍.3．（2023上·陜西·高三校聯(lián)考階段練習(xí)）已知函數(shù)SKIPIF1<0.(1)討論SKIPIF1<0的單調(diào)性；(2)若SKIPIF1<0在SKIPIF1<0上存2個(gè)零點(diǎn)，求SKIPIF1<0的取值范圍.4．（2023上·重慶涪陵·高三重慶市涪陵高級中學(xué)校?？奸_學(xué)考試）已知函數(shù)SKIPIF1<0.(1)若函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，求SKIPIF1<0的最小值；(2)若函數(shù)SKIPIF1<0的圖象與SKIPIF1<0有且只有一個(gè)交點(diǎn)，求SKIPIF1<0的取值范圍．5．（2023下·遼寧·高二校聯(lián)考期末）已知函數(shù)SKIPIF1<0，SKIPIF1<0，其中SKIPIF1<0，若SKIPIF1<0.(1)當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0的單調(diào)區(qū)間；(2)曲線SKIPIF1<0與直線SKIPIF1<0有且僅有兩個(gè)交點(diǎn)，求SKIPIF1<0的取值范圍.6．（2023下·廣東東莞·高二統(tǒng)考期末）已知函數(shù)SKIPIF1<0．(1)求函數(shù)SKIPIF1<0在SKIPIF1<0處的切線方程；(2)若SKIPIF1<0是SKIPIF1<0的極值點(diǎn)，且方程SKIPIF1<0有3個(gè)不同的實(shí)數(shù)解，求實(shí)數(shù)SKIPIF1<0的取值范圍．題型03利用數(shù)形結(jié)合法研究函數(shù)的零點(diǎn)（方程的根）問題1．（2023上·重慶涪陵·高三重慶市涪陵高級中學(xué)校?？奸_學(xué)考試）已知函數(shù)SKIPIF1<0.(1)若函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，求SKIPIF1<0的最小值；(2)若函數(shù)SKIPIF1<0的圖象與SKIPIF1<0有且只有一個(gè)交點(diǎn)，求SKIPIF1<0的取值范圍．2．（2023上·云南·高三校聯(lián)考階段練習(xí)）已知SKIPIF1<0．(1)當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0在SKIPIF1<0上的單調(diào)性；(2)若SKIPIF1<0，令SKIPIF1<0，討論方程SKIPIF1<0的解的個(gè)數(shù)．3．（2023下·湖南衡陽·高二?？茧A段練習(xí)）已知函數(shù)SKIPIF1<0，SKIPIF1<0其中SKIPIF1<0．(1)討論函數(shù)SKIPIF1<0的單調(diào)性；(2)若方程SKIPIF1<0有三個(gè)根，求SKIPIF1<0的取值范圍．

4．（2023上·廣東江門·高三統(tǒng)考階段練習(xí)）已知函數(shù)SKIPIF1<0.(1)求SKIPIF1<0的極值：(2)若SKIPIF1<0有兩個(gè)零點(diǎn)，求a的取值范圍.5．（2023上·北京·高三北京二十中?？茧A段練習(xí)）已知函數(shù)SKIPIF1<0，函數(shù)SKIPIF1<0，(1)已知直線SKIPIF1<0是曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線，且SKIPIF1<0與曲線SKIPIF1<0相切，求SKIPIF1<0的值；(2)若方程SKIPIF1<0有三個(gè)不同實(shí)數(shù)解，求實(shí)數(shù)SKIPIF1<0的取值范圍.6．（2023·四川·校聯(lián)考一模）已知函數(shù)SKIPIF1<0.(1)求SKIPIF1<0的單調(diào)區(qū)間；(2)令SKIPIF1<0（a為常數(shù)），若SKIPIF1<0有兩個(gè)零點(diǎn)SKIPIF1<0，求實(shí)數(shù)a的取值范圍.題型04構(gòu)造函數(shù)研究函數(shù)零點(diǎn)（方程的根）問題1．（2023·全國·模擬預(yù)測）已知函數(shù)SKIPIF1<0．(1)討論函數(shù)SKIPIF1<0的單調(diào)性．(2)若關(guān)于SKIPIF1<0的方程SKIPIF1<0有兩個(gè)實(shí)數(shù)根，求實(shí)數(shù)SKIPIF1<0的取值范圍．2．（2023上·湖南·高三邵陽市第二中學(xué)校聯(lián)考階段練習(xí)）已知函數(shù)SKIPIF1<0.(1)討論函數(shù)SKIPIF1<0的零點(diǎn)個(gè)數(shù)；(2)已知函數(shù)SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，關(guān)于SKIPIF1<0的方程SKIPIF1<0有兩個(gè)實(shí)根SKIPIF1<0，求證：SKIPIF1<0.（注：SKIPIF1<0是自然對數(shù)的底數(shù)）3．（2023上·重慶