專題20極值點偏移問題1．極值點偏移的含義若單峰函數(shù)f(x)的極值點為x0，則極值點的偏移問題的圖示及函數(shù)值的大小關(guān)系如下表所示.極值點x0函數(shù)值的大小關(guān)系圖示極值點不偏移x0＝eq\f(x1＋x2,2)f(x1)＝f(2x0－x2)極值點偏移左移x0<eq\f(x1＋x2,2)峰口向上：f(x1)<f(2x0－x2)峰口向下：f(x1)>f(2x0－x2)右移x0>eq\f(x1＋x2,2)峰口向上：f(x1)>f(2x0－x2)峰口向下：f(x1)<f(2x0－x2)2.函數(shù)極值點偏移問題的題型及解法極值點偏移問題的題設(shè)一般有以下四種形式：若函數(shù)f(x)在定義域上存在兩個零點x1，x2(x1≠x2)，求證：x1＋x2>2x0(x0為函數(shù)f(x)的極值點)；若在函數(shù)f(x)的定義域上存在x1，x2(x1≠x2)滿足f(x1)＝f(x2)，求證：x1＋x2>2x0(x0為函數(shù)f(x)的極值點)；(3)若函數(shù)f(x)存在兩個零點x1，x2(x1≠x2)，令x0＝eq\f(x1＋x2,2)，求證：f′(x0)>0；(4)若在函數(shù)f(x)的定義域上存在x1，x2(x1≠x2)滿足f(x1)＝f(x2)，令x0＝eq\f(x1＋x2,2)，求證：f′(x0)>0.3.極值點偏移問題的一般解法3.1對稱化構(gòu)造法主要用來解決與兩個極值點之和，積相關(guān)的不等式的證明問題．其解題要點如下：(1)定函數(shù)(極值點為SKIPIF1<0)，即利用導(dǎo)函數(shù)符號的變化判斷函數(shù)的單調(diào)性，進(jìn)而確定函數(shù)的極值點SKIPIF1<0．(2)構(gòu)造函數(shù)，即對結(jié)論SKIPIF1<0型，構(gòu)造函數(shù)SKIPIF1<0或SKIPIF1<0；(3)對結(jié)論SKIPIF1<0型，構(gòu)造函數(shù)SKIPIF1<0，通過研究SKIPIF1<0的單調(diào)性獲得不等式．(4)判斷單調(diào)性，即利用導(dǎo)數(shù)討論SKIPIF1<0的單調(diào)性．(5)比較大小，即判斷函數(shù)SKIPIF1<0在某段區(qū)間上的正負(fù)，并得出SKIPIF1<0與SKIPIF1<0的大小關(guān)系．(6)轉(zhuǎn)化，即利用函數(shù)f(x)的單調(diào)性，將SKIPIF1<0與SKIPIF1<0的大小關(guān)系轉(zhuǎn)化為SKIPIF1<0與SKIPIF1<0之間的關(guān)系，進(jìn)而得到所證或所求．3.2．差值代換法（韋達(dá)定理代換令SKIPIF1<0.）差值換元的目的也是消參、減元，就是根據(jù)已知條件首先建立極值點之間的關(guān)系，然后利用兩個極值點之差作為變量，從而實現(xiàn)消參、減元的目的．設(shè)法用差值(一般用SKIPIF1<0表示)表示兩個極值點，即SKIPIF1<0，化為單變量的函數(shù)不等式，繼而將所求解問題轉(zhuǎn)化為關(guān)于SKIPIF1<0的函數(shù)問題求解．3.3．比值代換法比值換元的目的也是消參、減元，就是根據(jù)已知條件首先建立極值點之間的關(guān)系，然后利用兩個極值點的比值作為變量，從而實現(xiàn)消參、減元的目的．設(shè)法用比值(一般用SKIPIF1<0表示)表示兩個極值點，即SKIPIF1<0，化為單變量的函數(shù)不等式，繼而將所求解問題轉(zhuǎn)化為關(guān)于SKIPIF1<0的函數(shù)問題求解．3.4．對數(shù)均值不等式法兩個正數(shù)SKIPIF1<0和SKIPIF1<0的對數(shù)平均定義：SKIPIF1<0對數(shù)平均與算術(shù)平均、幾何平均的大小關(guān)系：SKIPIF1<0（此式記為對數(shù)平均不等式）取等條件：當(dāng)且僅當(dāng)SKIPIF1<0時，等號成立．3.5指數(shù)不等式法在對數(shù)均值不等式中，設(shè)SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0，根據(jù)對數(shù)均值不等式有如下關(guān)系：SKIPIF1<0專項突破練1．已知函數(shù)SKIPIF1<0．(1)求函數(shù)SKIPIF1<0的單調(diào)區(qū)間；(2)當(dāng)SKIPIF1<0時，證明：SKIPIF1<0．2．已知函數(shù)SKIPIF1<0.(1)若SKIPIF1<0是增函數(shù)，求實數(shù)a的取值范圍；(2)若SKIPIF1<0有兩個極值點SKIPIF1<0，SKIPIF1<0，證明：SKIPIF1<0.3．已知函數(shù)SKIPIF1<0.(1)求SKIPIF1<0的極大值；(2)設(shè)SKIPIF1<0、SKIPIF1<0是兩個不相等的正數(shù)，且SKIPIF1<0，證明：SKIPIF1<0.4．已知函數(shù)SKIPIF1<0(1)討論f(x)的單調(diào)性；(2)若SKIPIF1<0，且SKIPIF1<0，證明:SKIPIF1<0.5．已知函數(shù)SKIPIF1<0（SKIPIF1<0且SKIPIF1<0）．(1)SKIPIF1<0，求函數(shù)SKIPIF1<0在SKIPIF1<0處的切線方程．(2)討論函數(shù)SKIPIF1<0的單調(diào)性；(3)若函數(shù)SKIPIF1<0有兩個零點SKIPIF1<0SKIPIF1<0，且SKIPIF1<0，證明：SKIPIF1<0．6．已知函數(shù)SKIPIF1<0(1)求證：當(dāng)SKIPIF1<0時，SKIPIF1<0；(2)當(dāng)方程SKIPIF1<0有兩個不等實數(shù)根SKIPIF1<0時，求證：SKIPIF1<07．已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0，證明：SKIPIF1<0；(2)若SKIPIF1<0有兩個不同的零點SKIPIF1<0，求a的取值范圍，并證明：SKIPIF1<0．8．已知函數(shù)SKIPIF1<0．(1)求曲線SKIPIF1<0在點SKIPIF1<0處的切線方程；(2)若SKIPIF1<0（SKIPIF1<0為SKIPIF1<0的導(dǎo)函數(shù)），方程SKIPIF1<0有兩個不等實根SKIPIF1<0、SKIPIF1<0，求證：SKIPIF1<0．9．已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0，證明：SKIPIF1<0時，SKIPIF1<0；(2)若函數(shù)SKIPIF1<0恰有三個零點SKIPIF1<0，證明：SKIPIF1<0．10．已知函數(shù)SKIPIF1<0．(1)若函數(shù)SKIPIF1<0為增函數(shù)，求實數(shù)SKIPIF1<0的取值范圍；(2)若函數(shù)SKIPIF1<0有兩個極值點SKIPIF1<0、SKIPIF1<0．求證：SKIPIF1<0．11．已知函數(shù)SKIPIF1<0.(1)求函數(shù)SKIPIF1<0的單調(diào)區(qū)間；(2)若函數(shù)SKIPIF1<0的圖象與SKIPIF1<0的圖象交于SKIPIF1<0，SKIPIF1<0兩點，證明：SKIPIF1<0.12．已知函數(shù)SKIPIF1<0.(1)討論SKIPIF1<0的單調(diào)性.(2)若函數(shù)SKIPIF1<0有兩個零點SKIPIF1<0，且SKIPIF1<0，證明：SKIPIF1<0.13．已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0時，SKIPIF1<0，求SKIPIF1<0的取值范圍；(2)當(dāng)SKIPIF1<0時，方程SKIPIF1<0有兩個不相等的實數(shù)根SKIPIF1<0，證明：SKIPIF1<0．14．設(shè)函數(shù)SKIPIF1<0，已知直線SKIPIF1<0是曲線SKIPIF1<0的一條切線.(1)求SKIPIF1<0的值，并討論函數(shù)SKIPIF1<0的單調(diào)性；(2)若SKIPIF1<0，其中SKIPIF1<0，證明：SKIPIF1<0.15．已知函數(shù)SKIPIF1<0有兩個不同的零點SKIPIF1<0.(1)求實數(shù)SKIPIF1<0的取值范圍；(2)求證：SKIPIF1<0.16．已知SKIPIF1<0是實數(shù)，函數(shù)SKIPIF1<0．(1)討論SKIPIF1<0的單調(diào)性；(2)若SKIPIF1<0有兩個相異的零點SKIPIF1<0且SKIPIF1<0，求證：SKIPIF1<0．17．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在SKIPIF1<0上有兩個不相等的零點SKIPIF1<0，求證：SKIPIF1<0.18．已知函數(shù)SKIPIF1<0的導(dǎo)函數(shù)為SKIPIF1<0.(1)判斷SKIPIF1<0的單調(diào)性；(2)若關(guān)于SKIPIF1<0的方程SKIPIF1<0有兩個實數(shù)根SKIPIF1<0，SKIPIF1<0，求證：SKIPIF1<0.19．已知函數(shù)SKIPIF1<0.(1)設(shè)函數(shù)SKIPIF1<0，且SKIPIF1<0恒成立，求實數(shù)SKIPIF1<0的取值范圍；(2)求證：SKIPIF1<0；(3)設(shè)函數(shù)SKIPIF1<0的兩個零點SKIPIF1<0、SKIPIF1<0，求證：SKIPIF1<0.20．已知函數(shù)SKIPIF1<0.（1）求SKIPIF1<0的單調(diào)區(qū)間與極值.（2）設(shè)SKIPIF1<0，SKIPIF1<0為兩個不相等的正數(shù)，且SKIPIF1<0，證明：SKIPIF1<0.21．已知函數(shù)SKIPIF1<0，其中SKIPIF1<0為自然對數(shù)的底數(shù).（1）討論函數(shù)SKIPIF1<0的單調(diào)性；（2）若SKIPIF1<0，且SKIPIF1<0，證明：SKIPIF1<0.22．已知函數(shù)SKIPIF1<0，其中SKIPIF1<0．(1)若SKIPIF1<0，求SKIPIF1<0的極值：(2)令函數(shù)SKIPIF1<0，若存在SKIPIF1<0，SKIPIF1<0使得SKIPIF1<0，證明：SKIPIF1<0．23．已知函數(shù)SKIPIF1<0.(1)求SKIPIF1<0的單調(diào)區(qū)間(2)若SKIPIF1<0的極值點為SKIPIF1<0，且S