新高考數(shù)學(xué)一輪復(fù)習(xí) 導(dǎo)數(shù)專項(xiàng)重點(diǎn)難點(diǎn)突破專題11 利用導(dǎo)數(shù)證明不等式(原卷版)

專題11利用導(dǎo)數(shù)證明不等式考點(diǎn)一單變量不等式的證明1.待證不等式的兩邊含有同一個(gè)變量時(shí)，一般地，可以直接構(gòu)造“左減右”的函數(shù)，利用導(dǎo)數(shù)研究其單調(diào)性，借助所構(gòu)造函數(shù)的單調(diào)性即可得證．2.若直接求導(dǎo)比較復(fù)雜或無從下手時(shí)，可將待證式進(jìn)行變形，構(gòu)造兩個(gè)都便于求導(dǎo)的函數(shù)，從而找到可以傳遞的中間量，達(dá)到證明的目標(biāo)．3.導(dǎo)數(shù)的綜合應(yīng)用題中，最常見就是ex和lnx與其他代數(shù)式結(jié)合的難題，對(duì)于這類問題，可以先對(duì)ex和lnx進(jìn)行放縮，使問題簡化，便于化簡或判斷導(dǎo)數(shù)的正負(fù)．常見的放縮公式如下：(1)ex≥1＋x，當(dāng)且僅當(dāng)x＝0時(shí)取等號(hào)；(2)ex≥ex，當(dāng)且僅當(dāng)x＝1時(shí)取等號(hào)；(3)當(dāng)x≥0時(shí)，ex≥1＋x＋eq\f(1,2)x2,當(dāng)且僅當(dāng)x＝0時(shí)取等號(hào)；(4)當(dāng)x≥0時(shí)，ex≥eq\f(e,2)x2＋1,當(dāng)且僅當(dāng)x＝0時(shí)取等號(hào)；(5)eq\f(x－1,x)≤lnx≤x－1≤x2－x，當(dāng)且僅當(dāng)x＝1時(shí)取等號(hào)；(6)當(dāng)x≥1時(shí)，eq\f(2x－1,x＋1)≤lnx≤eq\f(x－1,\r(x))，當(dāng)且僅當(dāng)x＝1時(shí)取等號(hào)．考點(diǎn)二雙變量不等式的證明破解含雙參不等式的證明的關(guān)鍵一是轉(zhuǎn)化，即由已知條件入手，尋找雙參所滿足的關(guān)系式，并把含雙參的不等式轉(zhuǎn)化為含單參的不等式；二是巧構(gòu)造函數(shù)，再借用導(dǎo)數(shù)，判斷函數(shù)的單調(diào)性，從而求其最值；三是回歸雙參的不等式的證明，把所求的最值應(yīng)用到雙參不等式，即可證得結(jié)果．考點(diǎn)三證明與數(shù)列有關(guān)的不等式(1)證明此類問題時(shí)常根據(jù)已知的函數(shù)不等式，用關(guān)于正整數(shù)n的不等式替代函數(shù)不等式中的自變量．通過多次求和達(dá)到證明的目的．此類問題一般至少有兩問，已知的不等式常由第一問根據(jù)待證式的特征而得到．(2)已知函數(shù)式為指數(shù)不等式(或?qū)?shù)不等式)，而待證不等式為與對(duì)數(shù)有關(guān)的不等式(或與指數(shù)有關(guān)的不等式)，還要注意指、對(duì)數(shù)式的互化，如ex＞x＋1可化為ln(x＋1)＜x等．專項(xiàng)突破一單變量不等式的證明1．已知SKIPIF1<0，SKIPIF1<0，SKIPIF1<0．(1)當(dāng)SKIPIF1<0時(shí)，求函數(shù)SKIPIF1<0的極值；(2)當(dāng)SKIPIF1<0時(shí)，求證：SKIPIF1<0．2．已知函數(shù)SKIPIF1<0.(1)若SKIPIF1<0在SKIPIF1<0上有2個(gè)零點(diǎn)，求a的取值范圍；(2)證明：SKIPIF1<0.3．已知函數(shù)SKIPIF1<0，SKIPIF1<0．(1)求曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程；(2)證明：SKIPIF1<0．4．已知函數(shù)SKIPIF1<0.（1）若函數(shù)SKIPIF1<0在定義域內(nèi)為增函數(shù)，求實(shí)數(shù)SKIPIF1<0的取值范圍；（2）若SKIPIF1<0且SKIPIF1<0，求證：SKIPIF1<0.5．已知函數(shù)SKIPIF1<0.(1)當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，求實(shí)數(shù)SKIPIF1<0的取值范圍；(2)證明：SKIPIF1<0.6．已知函數(shù)SKIPIF1<0.(1)若SKIPIF1<0有兩個(gè)極值點(diǎn)，求實(shí)數(shù)a的取值范圍；(2)當(dāng)SKIPIF1<0時(shí)，證明：SKIPIF1<0.7．已知函數(shù)SKIPIF1<0的最小值為SKIPIF1<0．(1)求實(shí)數(shù)SKIPIF1<0的值；(2)求證：當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0．8．已知函數(shù)SKIPIF1<0.(1)討論函數(shù)SKIPIF1<0的單調(diào)性；(2)當(dāng)SKIPIF1<0時(shí)，證明：SKIPIF1<0.9．已知函數(shù)SKIPIF1<0．(1)當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程；(2)當(dāng)SKIPIF1<0時(shí)，證明：SKIPIF1<0（其中SKIPIF1<0為自然對(duì)數(shù)的底數(shù)）．10．已知函數(shù)SKIPIF1<0.(1)求SKIPIF1<0的單調(diào)區(qū)間；(2)證明：SKIPIF1<0.11．已知函數(shù)SKIPIF1<0，SKIPIF1<0．(1)討論函數(shù)SKIPIF1<0的單調(diào)性，并求函數(shù)SKIPIF1<0的極值；(2)證明：對(duì)任意SKIPIF1<0，都有SKIPIF1<0．12．已知函數(shù)SKIPIF1<0，SKIPIF1<0(1)討論函數(shù)SKIPIF1<0的單調(diào)性；(2)當(dāng)SKIPIF1<0時(shí)，求證：SKIPIF1<0.13．已知函數(shù)SKIPIF1<0．(1)求曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程；(2)設(shè)SKIPIF1<0，討論函數(shù)SKIPIF1<0在SKIPIF1<0上的單調(diào)性；(3)證明：對(duì)任意的SKIPIF1<0，有SKIPIF1<0．14．已知函數(shù)SKIPIF1<0．(1)討論函數(shù)SKIPIF1<0的單調(diào)性；(2)若SKIPIF1<0，證明：SKIPIF1<0．專項(xiàng)突破二雙變量不等式的證明1．已知函數(shù)SKIPIF1<0，（SKIPIF1<0）．(1)若SKIPIF1<0存在兩個(gè)極值點(diǎn)，求實(shí)數(shù)SKIPIF1<0的取值范圍；(2)若SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的兩個(gè)極值點(diǎn)，證明：SKIPIF1<0．2．已知函數(shù)SKIPIF1<0.(1)若SKIPIF1<0在SKIPIF1<0上為單調(diào)函數(shù)，求實(shí)數(shù)a的取值范圍；(2)記SKIPIF1<0的兩個(gè)極值點(diǎn)為SKIPIF1<0，SKIPIF1<0，求證：SKIPIF1<0.3．設(shè)函數(shù)SKIPIF1<0(1)當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0的單調(diào)區(qū)間；(2)任意正實(shí)數(shù)SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，試判斷SKIPIF1<0與SKIPIF1<0的大小關(guān)系并證明4．記函數(shù)SKIPIF1<0，其導(dǎo)函數(shù)為SKIPIF1<0.(1)討論SKIPIF1<0的極值點(diǎn)個(gè)數(shù)；(2)當(dāng)SKIPIF1<0時(shí)，令SKIPIF1<0，若SKIPIF1<0是關(guān)于SKIPIF1<0的方程SKIPIF1<0的兩個(gè)相異的實(shí)數(shù)根，證明：SKIPIF1<0.5．已知函數(shù)SKIPIF1<0，且SKIPIF1<0．(1)求曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程；(2)若函數(shù)SKIPIF1<0有三個(gè)極值點(diǎn)SKIPIF1<0，且SKIPIF1<0，求證：SKIPIF1<0．6．已知函數(shù)SKIPIF1<0(1)當(dāng)SKIPIF1<0時(shí)，若對(duì)任意的SKIPIF1<0都有SKIPIF1<0求m的最大值(2)若函數(shù)SKIPIF1<0有且只有兩個(gè)不同的零點(diǎn)SKIPIF1<0求證SKIPIF1<07．已知函數(shù)SKIPIF1<0有兩個(gè)零點(diǎn)．(1)求a的取值范圍；(2)設(shè)SKIPIF1<0是SKIPIF1<0的兩個(gè)零點(diǎn)，證明：SKIPIF1<0．8．已知函數(shù)SKIPIF1<0.(1)若SKIPIF1<0，曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程為SKIPIF1<0，求實(shí)數(shù)SKIPIF1<0的值；(2)若SKIPIF1<0，SKIPIF1<0是SKIPIF1<0的兩個(gè)極值點(diǎn)，且SKIPIF1<0，證明：SKIPIF1<0.9．已知函數(shù)SKIPIF1<0，SKIPIF1<0．(1)求SKIPIF1<0的單調(diào)區(qū)間；(2)證明：SKIPIF1<0；(3)設(shè)a，b為正數(shù)，且SKIPIF1<0，證明：SKIPIF1<0．10．已知函數(shù)SKIPIF1<0．(1)討論函數(shù)SKIPIF1<0的單調(diào)區(qū)間；(2)當(dāng)SKIPIF1<0時(shí)，若SKIPIF1<0滿足SKIPIF1<0，求證：SKIPIF1<0．11．已知函數(shù)SKIPIF1<0.(1)求曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程；(2)設(shè)方程SKIPIF1<0的兩個(gè)根分別為SKIPIF1<0,SKIPIF1<0，證明：SKIPIF1<0.12．已知實(shí)數(shù)SKIPIF1<0，設(shè)函數(shù)SKIPIF1<0．(1)當(dāng)SKIPIF1<0時(shí)，求函數(shù)SKIPIF1<0的單調(diào)區(qū)間；(2)若函數(shù)SKIPIF1<0單調(diào)遞增，求a的最大值；(3)設(shè)SKIPIF1<0是SKIPIF1<0的兩個(gè)不同極值點(diǎn)，SKIPIF1<0是SKIPIF1<0的最大零點(diǎn)．證明：SKIPIF1<0．注：SKIPIF1<0是自然對(duì)數(shù)的底數(shù)．13．已知函數(shù)SKIPIF1<0．(1)若直線SKIPIF1<0與SKIPIF1<0的圖像相切，且切點(diǎn)的橫坐標(biāo)為1，求實(shí)數(shù)m和b的值；(2)若函數(shù)SKIPIF1<0在SKIPIF1<0上存在兩個(gè)極值點(diǎn)SKIPIF1<0，且SKIPIF1<0，證明：SKIPIF1<0．14．已知函數(shù)SKIPIF1<0(1)求函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線的方程；(2)若SKIPIF1<0有兩個(gè)極值點(diǎn)m，n，證明：SKIPIF1<0．專項(xiàng)突破三證明與數(shù)列有關(guān)的不等式1．已知關(guān)于SKIPIF1<0的函數(shù)SKIPIF1<0（1）討論SKIPIF1<0的單調(diào)性；（2）證明：當(dāng)SKIPIF1<0時(shí)，SKIPIF1<02．設(shè)函數(shù)SKIPIF1<0(1)求曲線SKIPIF1<0在SKIPIF1<0處的切線方程；(2)證明：當(dāng)SKIPIF1<0且SKIPIF1<0時(shí)，SKIPIF1<0.3．已知函數(shù)SKIPIF1<0．(1)若函數(shù)SKIPIF1<0在SKIPIF1<0處取得極值，求實(shí)數(shù)SKIPIF1<0的值，并求函數(shù)SKIPIF1<0的極值；(2)①若當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0恒成立，求實(shí)數(shù)SKIPIF1<0的取值范圍；②證明：當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0．4．已知函數(shù)f(x)＝lnx﹣ax+1（a∈R）．(1)求函數(shù)f(x)在區(qū)間[SKIPIF1<0