人教A版高中數(shù)學(xué)(必修第一冊)同步講義第27講 4.5.1函數(shù)的零點與方程的解（原卷版）

第05講4.5.1函數(shù)的零點與方程的解課程標(biāo)準(zhǔn)學(xué)習(xí)目標(biāo)①了解函數(shù)的零點與方程的解的關(guān)系，并能結(jié)合函數(shù)的圖象判定函數(shù)的零點。②能根據(jù)函數(shù)零點存在性定理對函數(shù)零點存在進行判定，同時能處理與函數(shù)零點問題相結(jié)合的求參數(shù)及綜合類的問題。通過本節(jié)課的學(xué)習(xí)，要求能判定函數(shù)零點的存在，同時能解決與函數(shù)零點相結(jié)合的綜合問題知識點01：函數(shù)零點的概念1、函數(shù)零點的概念對于一般函數(shù)SKIPIF1<0，我們把使SKIPIF1<0的實數(shù)SKIPIF1<0叫做函數(shù)SKIPIF1<0的零點.幾何定義:函數(shù)SKIPIF1<0的零點就是方程SKIPIF1<0的實數(shù)解,也就是函數(shù)SKIPIF1<0的圖象與SKIPIF1<0軸的公共點的橫坐標(biāo).

這樣：方程SKIPIF1<0有實數(shù)解SKIPIF1<0函數(shù)SKIPIF1<0有零點SKIPIF1<0函數(shù)SKIPIF1<0的圖象與SKIPIF1<0軸有公共點2、已學(xué)基本初等函數(shù)的零點①一次函數(shù)SKIPIF1<0只有一個零點SKIPIF1<0；②反比例函數(shù)SKIPIF1<0沒有零點；③指數(shù)函數(shù)SKIPIF1<0（SKIPIF1<0且SKIPIF1<0）沒有零點；④對數(shù)函數(shù)SKIPIF1<0（SKIPIF1<0且SKIPIF1<0）只有一個零點1；⑤冪函數(shù)SKIPIF1<0當(dāng)SKIPIF1<0時，有一個零點0；當(dāng)SKIPIF1<0時，無零點。知識點02：函數(shù)零點存在定理及其應(yīng)用1、函數(shù)零點存在定理如果函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的圖象是一條連續(xù)不斷的曲線，且有SKIPIF1<0，那么函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)至少有一個零點，即存在SKIPIF1<0，使得SKIPIF1<0，這個SKIPIF1<0也就是方程SKIPIF1<0的解.說明：定理要求具備兩個條件:①函數(shù)在區(qū)間SKIPIF1<0上的圖象是連續(xù)不斷的;②SKIPIF1<0.兩個條件缺一不可.2、函數(shù)零點的求法①代數(shù)法：根據(jù)零點定義，求出方程SKIPIF1<0的實數(shù)解;②數(shù)形結(jié)合法：作出函數(shù)圖象，利用函數(shù)性質(zhì)求解【即學(xué)即練1】（2023春·四川廣安·高一?？茧A段練習(xí)）函數(shù)SKIPIF1<0的零點為．【答案】2【詳解】令SKIPIF1<0，則SKIPIF1<0，得SKIPIF1<0.故答案為：SKIPIF1<03、函數(shù)零點個數(shù)的判斷①利用代數(shù)法，求出所有零點；②數(shù)形結(jié)合，通過作圖，找出圖象與SKIPIF1<0軸交點的個數(shù)；③數(shù)形結(jié)合，通過分離，將原函數(shù)拆分成兩個函數(shù)，找到兩個函數(shù)圖象交點的個數(shù)；④函數(shù)零點唯一：函數(shù)存在零點+函數(shù)單調(diào).知識點03：二次函數(shù)的零點問題一元二次方程SKIPIF1<0的實數(shù)根也稱為函數(shù)SKIPIF1<0的零點.當(dāng)SKIPIF1<0時，一元二次方程SKIPIF1<0的實數(shù)根、二次函數(shù)SKIPIF1<0的零點之間的關(guān)系如下表所示：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0的實數(shù)根SKIPIF1<0（其中SKIPIF1<0）SKIPIF1<0方程無實數(shù)根SKIPIF1<0的圖象SKIPIF1<0的零點SKIPIF1<0SKIPIF1<0函數(shù)無零點【即學(xué)即練2】（2023·高一課時練習(xí)）若函數(shù)SKIPIF1<0的一個零點是1，則它的另一個零點是．【答案】3【詳解】由SKIPIF1<0，所以令SKIPIF1<0或SKIPIF1<0，故另一個零點為3故答案為：3題型01求函數(shù)的零點【典例1】（2023·全國·高三專題練習(xí)）函數(shù)SKIPIF1<0的零點為．【典例2】（2023秋·遼寧鐵嶺·高一鐵嶺市清河高級中學(xué)?？计谀┮阎瘮?shù)SKIPIF1<0，則函數(shù)SKIPIF1<0的零點為.【變式1】（2023春·浙江·高一校聯(lián)考期中）函數(shù)SKIPIF1<0的零點是【變式2】（2023·江蘇·高一假期作業(yè)）求下列函數(shù)的零點．(1)SKIPIF1<0；(2)SKIPIF1<0.題型02函數(shù)零點個數(shù)的判斷【典例1】（2023·全國·高一假期作業(yè)）函數(shù)SKIPIF1<0的零點個數(shù)為（）A．1 B．2C．1或2 D．0【典例2】（2023·高一課時練習(xí)）方程SKIPIF1<0的實數(shù)解的個數(shù)是（

）A．0 B．1 C．2 D．3【典例3】（2023·全國·高三專題練習(xí)）已知SKIPIF1<0，方程SKIPIF1<0的實根個數(shù)為．【典例4】（2023春·山東德州·高二?？茧A段練習(xí)）若函數(shù)SKIPIF1<0，則函數(shù)SKIPIF1<0的零點個數(shù)是.【變式1】（2023·全國·高一假期作業(yè)）函數(shù)SKIPIF1<0的零點的個數(shù)是（

）A．0 B．1 C．2 D．無數(shù)個【變式2】（2023·江蘇·高一假期作業(yè)）已知函數(shù)SKIPIF1<0.(1)作出函數(shù)SKIPIF1<0的圖象；(2)就a的取值范圍討論函數(shù)SKIPIF1<0的零點的個數(shù)．【變式3】（2023·上海浦東新·華師大二附中?？寄M預(yù)測）若SKIPIF1<0的值域為SKIPIF1<0，則SKIPIF1<0至多有個零點.【變式4】（2023·全國·高三對口高考）函數(shù)SKIPIF1<0的圖象與函數(shù)SKIPIF1<0的圖象的交點個數(shù)為個．題型03判斷函數(shù)零點所在的區(qū)間【典例1】（2023春·云南楚雄·高一統(tǒng)考期末）若SKIPIF1<0是方程SKIPIF1<0的解，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023春·天津紅橋·高二統(tǒng)考學(xué)業(yè)考試）設(shè)SKIPIF1<0為方程SKIPIF1<0的解，若SKIPIF1<0，則SKIPIF1<0的值為.【變式1】（2023秋·重慶長壽·高一統(tǒng)考期末）函數(shù)SKIPIF1<0的零點所在區(qū)間是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】（2023春·湖南岳陽·高一湖南省岳陽縣第一中學(xué)?？计谀┖瘮?shù)SKIPIF1<0的零點為SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，則k的值為（

）A．1 B．2 C．0 D．3題型04已知零點個數(shù)求參數(shù)的取值范圍【典例1】（2023·全國·高一假期作業(yè)）若方程SKIPIF1<0有兩個不同的實數(shù)根，則實數(shù)SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023·湖南常德·常德市一中校考模擬預(yù)測）設(shè)SKIPIF1<0表示m，n中的較小數(shù)．若函數(shù)SKIPIF1<0至少有3個零點，則實數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例3】（2023·高一課時練習(xí)）若函數(shù)SKIPIF1<0有2個零點，求實數(shù)a的取值范圍．【典例4】（2023春·云南昆明·高三云南省昆明市第十二中學(xué)?？茧A段練習(xí)）已知函數(shù)SKIPIF1<0是偶函數(shù).當(dāng)SKIPIF1<0時，SKIPIF1<0.(1)求函數(shù)SKIPIF1<0在SKIPIF1<0上的解析式；(2)若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)，求實數(shù)a的取值范圍；(3)已知SKIPIF1<0，試討論SKIPIF1<0的零點個數(shù)，并求對應(yīng)的m的取值范圍.【變式1】（2023·北京·高三專題練習(xí)）設(shè)SKIPIF1<0，函數(shù)SKIPIF1<0若SKIPIF1<0恰有一個零點，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式2】（多選）（2023·全國·高三專題練習(xí)）已知函數(shù)SKIPIF1<0，若關(guān)于x的方程SKIPIF1<0恰有兩個互異的實數(shù)解，則實數(shù)a的值可以是（

）A．0 B．1 C．SKIPIF1<0 D．2【變式3】（2023·全國·高一假期作業(yè)）若函數(shù)SKIPIF1<0在SKIPIF1<0內(nèi)有且只有一個零點，則SKIPIF1<0的取值集合是.【變式4】（2023·高一課時練習(xí)）已知SKIPIF1<0是定義在SKIPIF1<0上的偶函數(shù)．(1)求SKIPIF1<0的值；(2)畫出SKIPIF1<0的圖象，并指出其單調(diào)減區(qū)間；(3)若關(guān)于SKIPIF1<0的方程SKIPIF1<0有2個不相等的實數(shù)根，求實數(shù)SKIPIF1<0的取值范圍．題型05已知零點所在區(qū)間求參數(shù)的取值范圍【典例1】（2023春·河南信陽·高一統(tǒng)考期末）函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上存在零點，則實數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】（多選）（2023秋·高一單元測試）函數(shù)SKIPIF1<0的一個零點在區(qū)間SKIPIF1<0內(nèi)，則實數(shù)a的可能取值是（

）A．0 B．1 C．2 D．3【典例3】（2023·全國·高三專題練習(xí)）設(shè)SKIPIF1<0為實數(shù)，函數(shù)SKIPIF1<0在SKIPIF1<0上有零點，則實數(shù)SKIPIF1<0的取值范圍為．【變式1】（2023·高一課時練習(xí)）若函數(shù)SKIPIF1<0在SKIPIF1<0內(nèi)恰有一個零點，則a的取值范圍（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】（2023春·上海青浦·高一統(tǒng)考開學(xué)考試）若關(guān)于SKIPIF1<0的方程SKIPIF1<0在SKIPIF1<0上有解，則實數(shù)SKIPIF1<0的取值范圍是．【變式3】（2023秋·湖北襄陽·高一統(tǒng)考期末）若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)有零點，則實數(shù)SKIPIF1<0的取值范圍是.題型06二次函數(shù)的零點問題【典例1】（2023秋·江蘇常州·高一常州市北郊高級中學(xué)?？计谀┮阎瘮?shù)SKIPIF1<0的零點為SKIPIF1<0，滿足SKIPIF1<0，則SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023·高一課時練習(xí)）方程SKIPIF1<0的一根大于1，一根小于1，則實數(shù)SKIPIF1<0的取值范圍是.【典例3】（2023·江蘇·高一假期作業(yè)）（1）判斷二次函數(shù)SKIPIF1<0在SKIPIF1<0內(nèi)是否存在零點；（2）若二次函數(shù)SKIPIF1<0SKIPIF1<0的兩個零點均為正數(shù)，求實數(shù)SKIPIF1<0的取值范圍．【變式1】（2023·云南紅河·彌勒市一中校考模擬預(yù)測）已知關(guān)于SKIPIF1<0的方程SKIPIF1<0，SKIPIF1<0存在兩個不同的實根，則實數(shù)SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式2】（2023·高一課時練習(xí)）若關(guān)于SKIPIF1<0的方程SKIPIF1<0在SKIPIF1<0內(nèi)有解，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3】（2023·江蘇·高一假期作業(yè)）已知函數(shù)SKIPIF1<0SKIPIF1<0.(1)若該函數(shù)有兩個不相等的正零點，求SKIPIF1<0的取值范圍；(2)若該函數(shù)有兩個零點，一個大于1，另一個小于1，求SKIPIF1<0的取值范圍．題型07函數(shù)與方程綜合【典例1】（2023秋·高一單元測試）已知函數(shù)SKIPIF1<0，常數(shù)SKIPIF1<0.(1)若SKIPIF1<0是奇函數(shù)，求SKIPIF1<0的值；(2)若SKIPIF1<0，SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)有且僅有一個零點，求實數(shù)SKIPIF1<0的取值范圍.【典例2】（2023春·福建福州·高二福建省福州延安中學(xué)?？紝W(xué)業(yè)考試）已知函數(shù)SKIPIF1<0(1)證明：函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞減；(2)討論關(guān)于x的方程SKIPIF1<0的實數(shù)解的個數(shù)．【變式1】（2023秋·江蘇無錫·高一無錫市第一中學(xué)?？计谀┮阎瘮?shù)SKIPIF1<0為奇函數(shù).(1)求實數(shù)a的值；(2)若方程SKIPIF1<0在區(qū)間SKIPIF1<0上無解，求實數(shù)m的取值范圍.【變式2】（2023·全國·高三專題練習(xí)）已知函數(shù)SKIPIF1<0是定義在SKIPIF1<0上的偶函數(shù)，且當(dāng)SKIPIF1<0時，SKIPIF1<0，函數(shù)SKIPIF1<0在SKIPIF1<0軸左側(cè)的圖象如圖所示．(1)求函數(shù)SKIPIF1<0的解析式；(2)若關(guān)于SKIPIF1<0的方程SKIPIF1<0有SKIPIF1<0個不相等的實數(shù)根，求實數(shù)SKIPIF1<0的取值范圍．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春·黑龍江齊齊哈爾·高一校聯(lián)考開學(xué)考試）已知函數(shù)SKIPIF1<0，則SKIPIF1<0的零點所在的區(qū)間為（

）.A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2023·全國·高一假期作業(yè)）已知方程SKIPIF1<0的解在SKIPIF1<0內(nèi)，則SKIPIF1<0（

）A．3 B．2 C．1 D．03．（2023·江蘇·高一假期作業(yè)）已知函數(shù)SKIPIF1<0的兩個零點都大于2，則實數(shù)m的取值范圍是（）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<04．（2023·高一課時練習(xí)）已知二次函數(shù)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)的零點情況是（

）A．有兩個零點 B．有唯一零點 C．沒有零點 D．不確定5．（2023春·山東聊城·高二統(tǒng)考期末）已知函數(shù)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0的零點分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<06．（2023春·浙江溫州·高二統(tǒng)考學(xué)業(yè)考試）設(shè)實數(shù)a為常數(shù)，則函數(shù)SKIPIF1<0存在零點的充分必要條件是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．（2023春·江蘇南通·高二統(tǒng)考期末）已知函數(shù)SKIPIF1<0若函數(shù)SKIPIF1<0有五個零點，則實數(shù)SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．（2023春·河南駐馬店·高一河南省駐馬店高級中學(xué)?？茧A段練習(xí)）享有“數(shù)學(xué)王子”稱號的德國數(shù)學(xué)家高斯，是近代數(shù)學(xué)奠基者之一，SKIPIF1<0被稱為“高斯函數(shù)”，其中SKIPIF1<0表示不超過SKIPIF1<0的最大整數(shù)，例如：SKIPIF1<0，設(shè)SKIPIF1<0為函數(shù)SKIPIF1<0的零點，則SKIPIF1<0（

）A．3 B．4 C．5 D．6二、多選題9．（2023·全國·高一假期作業(yè)）函數(shù)SKIPIF1<0的零點可以是（）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<010．（2023·高一課時練習(xí)）設(shè)SKIPIF1<0為定義在R上的奇函數(shù)，當(dāng)SKIPIF1<0時，SKIPIF1<0為常數(shù)），則（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．函數(shù)SKIPIF1<0僅有一個零點三、填空題11．（2023·江蘇·高一假期作業(yè)）已知函數(shù)SKIPIF1<0在SKIPIF1<0上有零點，則實數(shù)a的取值范圍是．12．（2023春·江蘇揚州·高一揚州市廣陵區(qū)紅橋高級中學(xué)校考期中）若SKIPIF1<0是方程SKIPIF1<0的解，則SKIPIF1<0在區(qū)間內(nèi)(填序號).①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0；④SKIPIF1<0.四、解答題13．（2023·高一課時練習(xí)）已知一次函數(shù)SKIPIF1<0滿足SKIPIF1<0，SKIPIF1<0．（1）求這個函數(shù)的解析式；（2）若函數(shù)SKIPIF1<0，求函數(shù)SKIPIF1<0的零點．14．（2023秋·陜西西安·高一統(tǒng)考期末）已知函數(shù)SKIPIF1<0，且SKIPIF1<0的圖象經(jīng)過點SKIPIF1<0.(1)求SKIPIF1<0的值；(2)求SKIPIF1<0在區(qū)間SKIPIF1<0上的最小值；(3)若SKIPIF1<0，求證：SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)存在零點.B能力提升1．（2023春·浙江衢州·高一統(tǒng)考期末）已知函數(shù)SKIPIF1<0，若SKIPIF1<0且SKIPIF1<0，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2023春·天津·高二統(tǒng)考期末）已知函數(shù)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0．若SKIPIF1<0恰有2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．（2023春·浙江·高一景寧中學(xué)校聯(lián)考階段練習(xí)）函數(shù)SKIPIF1<0滿足：①SKIPIF1<0在SKIPIF1<0內(nèi)是單調(diào)遞增函數(shù)；②SKIPIF1<0在SKIPIF1<0上的值域為SKIPIF1<0，則稱區(qū)間SKIPIF1<0為SKIPIF1<0的SKIPIF1<0級“調(diào)和區(qū)間”.若函數(shù)SKIPIF1<0