新高考數(shù)學(xué)二輪復(fù)習(xí)鞏固練習(xí)13 導(dǎo)數(shù)解答題之雙變量問題（原卷版）

微專題13導(dǎo)數(shù)解答題之雙變量問題【秒殺總結(jié)】1、破解雙參數(shù)不等式的方法：一是轉(zhuǎn)化，即由已知條件入手，尋找雙參數(shù)滿足的關(guān)系式，并把含雙參數(shù)的不等式轉(zhuǎn)化為含單參數(shù)的不等式；二是巧構(gòu)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；四是主元法.【典型例題】例1．（2023·上海·高三專題練習(xí)）已知函數(shù)SKIPIF1<0，其中SKIPIF1<0為自然對數(shù)的底數(shù)，約為SKIPIF1<0.(1)求函數(shù)SKIPIF1<0的極小值；(2)若實數(shù)SKIPIF1<0滿足SKIPIF1<0且SKIPIF1<0，證明：SKIPIF1<0.例2．（2023·上海·高三專題練習(xí)）已知函數(shù)SKIPIF1<0.(1)當SKIPIF1<0時，求函數(shù)SKIPIF1<0在點SKIPIF1<0處的切線方程；(2)當SKIPIF1<0，求函數(shù)SKIPIF1<0的最大值；(3)若函數(shù)SKIPIF1<0在定義域內(nèi)有兩個不相等的零點SKIPIF1<0，證明：SKIPIF1<0.例3．（2023·吉林長春·高三長春市第二中學(xué)校考期末）已知函數(shù)SKIPIF1<0．(1)當SKIPIF1<0時，求SKIPIF1<0在SKIPIF1<0處的切線方程；(2)當SKIPIF1<0時，SKIPIF1<0恒成立，求SKIPIF1<0的取值范圍；(3)證明：當SKIPIF1<0時，SKIPIF1<0．例4．（2023·廣西柳州·統(tǒng)考模擬預(yù)測）已知SKIPIF1<0，記SKIPIF1<0的導(dǎo)函數(shù)為SKIPIF1<0．(1)討論SKIPIF1<0的單調(diào)性；(2)若SKIPIF1<0有三個零點SKIPIF1<0，且SKIPIF1<0，證明：SKIPIF1<0．例5．（2023·浙江·高三校聯(lián)考期末）已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0，求SKIPIF1<0的值；(2)證明：SKIPIF1<0．例6．（2023·湖北·宜昌市一中校聯(lián)考模擬預(yù)測）已知直線l與曲線SKIPIF1<0相切于點SKIPIF1<0．證明：(1)l與曲線SKIPIF1<0恰存在兩個公共點SKIPIF1<0；(2)SKIPIF1<0．例7．（2023·浙江麗水·高三浙江省麗水中學(xué)校聯(lián)考期末）已知函數(shù)SKIPIF1<0.(1)求函數(shù)SKIPIF1<0在點SKIPIF1<0處的切線方程；(2)若SKIPIF1<0為方程SKIPIF1<0的兩個不相等的實根，證明：（i）SKIPIF1<0；（ii）SKIPIF1<0.例8．（2023·河北邯鄲·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0（其中e為自然對數(shù)的底數(shù)）.(1)求曲線SKIPIF1<0在SKIPIF1<0處的切線方程；(2)已知SKIPIF1<0是SKIPIF1<0的極大值點，若SKIPIF1<0，且SKIPIF1<0.證明：SKIPIF1<0.例9．（2023·江蘇無錫·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0有兩個零點，求a的取值范圍；(2)若方程SKIPIF1<0有兩個實數(shù)根SKIPIF1<0，且SKIPIF1<0，證明：SKIPIF1<0．【過關(guān)測試】1．（2023·浙江紹興·高三期末）已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0，記SKIPIF1<0的最小值為m，求證：SKIPIF1<0．(2)方程SKIPIF1<0有兩個不同的實根SKIPIF1<0，且SKIPIF1<0，求證：SKIPIF1<0．2．（2023·浙江·高三期末）已知函數(shù)SKIPIF1<0．(1)證明：函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上有2個零點；(2)若函數(shù)SKIPIF1<0有兩個極值點：SKIPIF1<0，且SKIPIF1<0．求證：SKIPIF1<0（其中SKIPIF1<0為自然對數(shù)的底數(shù)）.3．（2023·河南三門峽·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0與函數(shù)SKIPIF1<0有相同的極值點與極值.(1)求a，b；(2)若方程SKIPIF1<0與SKIPIF1<0分別有兩個解p，q（SKIPIF1<0）和r，s（SKIPIF1<0）.①分別用p，q表示出r，s；②求證：SKIPIF1<0.4．（2023·河北石家莊·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0.(1)設(shè)SKIPIF1<0，若SKIPIF1<0在SKIPIF1<0上恒成立，求實數(shù)SKIPIF1<0的取值范圍；(2)設(shè)SKIPIF1<0，若存在正實數(shù)SKIPIF1<0，滿足SKIPIF1<0，證明：SKIPIF1<0.5．（2023·河北唐山·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0.(1)求SKIPIF1<0的極值；(2)若SKIPIF1<0，證明:函數(shù)SKIPIF1<0有兩個零點SKIPIF1<0，且SKIPIF1<0.6．（2023·江蘇揚州·高三校聯(lián)考期末）已知函數(shù)SKIPIF1<0，SKIPIF1<0．(1)若SKIPIF1<0的最值和SKIPIF1<0的最值相等，求m的值；(2)證明：若函數(shù)SKIPIF1<0有兩個零點SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0．7．（2023春·江蘇南京·高三南京師大附中?？奸_學(xué)考試）已知函數(shù)SKIPIF1<0，SKIPIF1<0為函數(shù)SKIPIF1<0的導(dǎo)函數(shù)(1)討論SKIPIF1<0的單調(diào)性；(2)當SKIPIF1<0時，SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，證明：SKIPIF1<0.8．（2023·江西吉安·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0，SKIPIF1<0.(1)當SKIPIF1<0時，求曲線SKIPIF1<0在SKIPIF1<0處的切線方程；(2)設(shè)SKIPIF1<0，SKIPIF1<0是SKIPIF1<0的兩個不同零點，證明：SKIPIF1<0.9．（2023·天津南開·高三崇化中學(xué)?？计谀┮阎瘮?shù)SKIPIF1<0．(1)若實數(shù)SKIPIF1<0，求函數(shù)SKIPIF1<0在點SKIPIF1<0處的切線方程；(2)討論函數(shù)SKIPIF1<0的單調(diào)性；(3)設(shè)SKIPIF1<0，若SKIPIF1<0且SKIPIF1<0，使得SKIPIF1<0，證明：SKIPIF1<0．10．（2023·江西·高三校聯(lián)考期末）已知函數(shù)SKIPIF1<0（SKIPIF1<0是自然對數(shù)的底數(shù)）有兩個零點.(1)求實數(shù)SKIPIF1<0的取值范圍：(2)若SKIPIF1<0的兩個零點分別為SKIPIF1<0，證明：SKIPIF1<011．（2023·全國·高三專題練習(xí)）已知函數(shù)SKIPIF1<0，SKIPIF1<0．(1)討論SKIPIF1<0的單調(diào)區(qū)間；(2)當SKIPIF1<0時，證明SKIPIF1<0．12．（2023·山東菏澤·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0，SKIPIF1<0．(1)證明：SKIPIF1<0存在唯一零點；(2)設(shè)SKIPIF1<0，若存在SKIPIF1<0，使得SKIPIF1<0，證明：SKIPIF1<0．13．（2023·全國·高三專題練習(xí)）設(shè)SKIPIF1<0，SKIPIF1<0.(1)設(shè)SKIPIF1<0，討論函數(shù)SKIPIF1<0的單調(diào)性；(2)若函數(shù)SKIPIF1<0在SKIPIF1<0有兩個零點SKIPIF1<0，SKIPIF1<0，證明：SKIPIF1<0.14．（2023·四川成都·高三樹德中學(xué)?？计谀┮阎瘮?shù)SKIPIF1<0.(1)討論SKIPIF1<0的單調(diào)性；(2)設(shè)SKIPIF1<0是兩個不相等的正數(shù)，且SKIPIF1<0，證明：SKIPIF1<0.15．（2023·安徽馬鞍山·統(tǒng)考一模）設(shè)函數(shù)SKIPIF1<0.(1)若SKIPIF1<0對SKIPIF1<0恒成立，求實數(shù)SKIPIF1<0的取值范圍；(2)已知方程SKIPIF1<0有兩個不同的根SKIPIF1<0、SKIPIF1<0，求證：SKIPIF1<0，其中SKIPIF1<0為自然對數(shù)的底數(shù).16．（2023·江蘇·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0(1)當SKIPIF1<0時，求SKIPIF1<0的單調(diào)區(qū)間；(2)若SKIPIF1<0有兩個零點SKIPIF1<0，求SKIPIF1<0的范圍，并證明SKIPIF1<017．（2023·全國·高三專題練習(xí)）已知函數(shù)SKIPIF1<0，設(shè)曲線SKIPIF1<0在點SKIPIF1<0處的切線方程為SKIPIF1<0.(1)證明：對定義域內(nèi)任意SKIPIF1<0，都有SKIPIF1<0；(2)當SKIPIF1<0時，關(guān)于SKIPIF1<0的方程SKIPIF1<0有兩個不等的實數(shù)根SKIPIF1<0，證明：SKIPIF1<0.18．（2023·全國·高二專題練習(xí)）已知函數(shù)SKIPIF1<0．(1)當SKIPIF1<0時，求曲線SKIPIF1<0在SKIPIF1<0處的切線方程；(2)設(shè)SKIPIF1<0，證明：對任意SKIPIF1<0，SKIPIF1<0，SKIPIF1<0．19．（2023·四川·校聯(lián)考一模）已知函數(shù)SKIPIF1<0.(1)求曲線SKIPIF1<0在點SKIPIF1<0處的切線方程；(2)設(shè)SKIPIF1<0，證明：SKIPIF1<0.20．（2023·四川瀘州·瀘州老窖天府中學(xué)校考模擬預(yù)測）設(shè)函數(shù)SKIPIF1<0.(1)討論SKIPIF1<0的單調(diào)性；(2)SKIPIF1<0時，若SKIPIF1<0，求證：SKIPIF1<0.21．（2023春·湖南長沙·高三長郡中學(xué)校考階段練習(xí)）已知函數(shù)SKIPIF1<0.(1)若SKIPIF1<0，求SKIPIF1<0的取值范圍；(2)記SKIPIF1<0的零點為SKIPIF1<0（SKIPIF1<0），SKIPIF1<0的極值點為SKIPIF1<0，證明：SKIPIF1<0.22．（2023·天津南開·高三南開中學(xué)?？茧A段練習(xí)）已知函數(shù)SKIPIF1<0，SKIPIF1<0為常數(shù)SKIPIF1<0，SKIPIF1<0(1)若函數(shù)SKIPIF1<0在原點的切線與函數(shù)SKIPIF1<0的圖象也相切，求b；(2)當SKIPIF1<0時，SKIPIF1<0，使SKIPIF1<0成立，求M的最大值；(3)若函數(shù)SKIPIF1<0的圖象與x軸有兩個不同的交點SKIPIF1<0，且SKIPIF1<0，證明：SKIPIF1<023．（2023·全國·高三專題練習(xí)）已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0單調(diào)遞增，求a的取值范圍；(2)若SKIPIF1<0有兩個極值點SKIPIF1<0，其中SKIPIF1<0，求證：SKIPIF1<0．24．（2023·全國·高三專題練習(xí)）已知函數(shù)SKIPIF1<0，曲線SKIPIF1<0在點SKIPIF1<0處的切線與直線SKIPIF1<0垂直.(1)試比較SKIPIF1<0與SKIPIF1<0的大小，并說明理由；(2)若函數(shù)SKIPIF1<0有兩個不同的零點SKIPIF1<0，證明：SKIPIF1<0.25．（2023·全國·高三專題練習(xí)）已知SKIPIF1<0為自然對數(shù)的底數(shù)SKIPIF1<0.(1)討論函數(shù)SKIPIF1<0的單調(diào)性；(2)若函數(shù)SKIPIF1<0有兩個不同零點SKIPIF1<0，求證：SKIPIF1