新高考數(shù)學二輪復習鞏固練習11 導數(shù)解答題之極最值問題（原卷版）

微專題11導數(shù)解答題之極最值問題【秒殺總結】1、利用導數(shù)求函數(shù)的極最值問題．解題方法是利用導函數(shù)與單調性關系確定單調區(qū)間，從而求得極最值．只是對含有參數(shù)的極最值問題，需要對導函數(shù)進行二次討論，對導函數(shù)或其中部分函數(shù)再一次求導，確定單調性，零點的存在性及唯一性等，由于零點的存在性與參數(shù)有關，因此對函數(shù)的極最值又需引入新函數(shù)，對新函數(shù)再用導數(shù)進行求值、證明等操作．【典型例題】例1．（2023秋·江蘇泰州·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0（SKIPIF1<0為非零常數(shù)），記SKIPIF1<0，SKIPIF1<0.(1)當SKIPIF1<0時，SKIPIF1<0恒成立，求實數(shù)SKIPIF1<0的最大值；(2)當SKIPIF1<0時，設SKIPIF1<0，對任意的SKIPIF1<0，當SKIPIF1<0時，SKIPIF1<0取得最小值，證明：SKIPIF1<0且所有點SKIPIF1<0在一條定直線上；(3)若函數(shù)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0都存在極小值，求實數(shù)SKIPIF1<0的取值范圍.例2．（2023秋·遼寧葫蘆島·高三葫蘆島第一高級中學?？计谀┮阎瘮?shù)SKIPIF1<0(1)若SKIPIF1<0，求曲線SKIPIF1<0在點SKIPIF1<0處的切線方程；(2)討論函數(shù)SKIPIF1<0的單調區(qū)間；(3)設SKIPIF1<0，若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上存在極值點，求SKIPIF1<0的取值范圍.例3．（2023秋·山東濰坊·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0.(1)若SKIPIF1<0，求證；函數(shù)SKIPIF1<0的圖象與SKIPIF1<0軸相切于原點；(2)若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0，SKIPIF1<0各恰有一個極值點，求實數(shù)SKIPIF1<0的取值范圍.例4．（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0.若函數(shù)SKIPIF1<0有兩個極值點SKIPIF1<0，且不等式SKIPIF1<0恒成立，試求實數(shù)SKIPIF1<0的取值范圍.例5．（2023春·江蘇南京·高三南京市第一中學校考開學考試）已知函數(shù)f（x）＝ae﹣x+lnx﹣1（a∈R）．(1)當a≤e時，討論函數(shù)f（x）的單調性：(2)若函數(shù)f（x）恰有兩個極值點x1，x2（x1＜x2），且x1+x2≤2ln3，求SKIPIF1<0的最大值．例6．（2023·重慶·統(tǒng)考一模）已知函數(shù)SKIPIF1<0，設SKIPIF1<0為SKIPIF1<0的導函數(shù)SKIPIF1<0．(1)討論SKIPIF1<0的零點個數(shù)；(2)當SKIPIF1<0時，記SKIPIF1<0的最小值為SKIPIF1<0，求SKIPIF1<0的最大值．例7．（2023·江西景德鎮(zhèn)·統(tǒng)考模擬預測）已知函數(shù)SKIPIF1<0.(1)若函數(shù)SKIPIF1<0在定義域上單調遞增,求SKIPIF1<0的最大值;(2)若函數(shù)SKIPIF1<0在定義域上有兩個極值點SKIPIF1<0和SKIPIF1<0,若SKIPIF1<0,求SKIPIF1<0的最大值.例8．（2023秋·山西運城·高三統(tǒng)考期末）已知SKIPIF1<0．(1)求證：SKIPIF1<0恒成立；(2)令SKIPIF1<0，討論SKIPIF1<0在SKIPIF1<0上的極值點個數(shù)．【過關測試】1．（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0，其中a為大于0的常數(shù)，若SKIPIF1<0．(1)討論SKIPIF1<0的單調區(qū)間；(2)若SKIPIF1<0在SKIPIF1<0取得極小值，求SKIPIF1<0的最小值．2．（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0，求函數(shù)SKIPIF1<0的所有零點；(2)若SKIPIF1<0，證明函數(shù)SKIPIF1<0不存在極值．3．（2023·四川內江·統(tǒng)考一模）已知函數(shù)SKIPIF1<0(1)當SKIPIF1<0時，求f（x）的單調遞增區(qū)間：(2)若函數(shù)f（x）恰有兩個極值點，記極大值和極小值分別為M、m，求證：SKIPIF1<0.4．（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0為常數(shù)SKIPIF1<0，且SKIPIF1<0在定義域內有兩個極值點.（1）求SKIPIF1<0的取值范圍；（2）設函數(shù)SKIPIF1<0的兩個極值點分別為SKIPIF1<0，求SKIPIF1<0的范圍.5．（2023·四川攀枝花·統(tǒng)考二模）已知函數(shù)SKIPIF1<0．(1)當SKIPIF1<0時，求函數(shù)SKIPIF1<0的極值；(2)設函數(shù)SKIPIF1<0，若SKIPIF1<0有兩個零點SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0為SKIPIF1<0的唯一極值點，求證：SKIPIF1<0．6．（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0，SKIPIF1<0．(1)討論函數(shù)SKIPIF1<0的單調性；(2)當SKIPIF1<0時，設SKIPIF1<0，SKIPIF1<0，函數(shù)SKIPIF1<0有兩個極值點SKIPIF1<0．①求m的取值范圍；②若SKIPIF1<0，求SKIPIF1<0的取值范圍．7．（2023秋·黑龍江哈爾濱·高三哈師大附中?？计谀┮阎瘮?shù)SKIPIF1<0有兩個極值點SKIPIF1<0，SKIPIF1<0.(1)求實數(shù)SKIPIF1<0的取值范圍；(2)求證：SKIPIF1<0.8．（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0.(1)若函數(shù)SKIPIF1<0有兩個極值（i）求實數(shù)SKIPIF1<0的取值范圍；（ii）求SKIPIF1<0極大值的取值范圍.(2)對于函數(shù)SKIPIF1<0，都有SKIPIF1<0，則稱SKIPIF1<0在區(qū)間SKIPIF1<0上是凸函數(shù).利用上述定義證明，當SKIPIF1<0時，SKIPIF1<0在SKIPIF1<0上是凸函數(shù).9．（2023秋·黑龍江綏化·高三?？计谀┮阎獙崝?shù)SKIPIF1<0，函數(shù)SKIPIF1<0，SKIPIF1<0是自然對數(shù)的底數(shù).(1)當SKIPIF1<0時，求函數(shù)SKIPIF1<0的單調區(qū)間；(2)求證：SKIPIF1<0存在極值點SKIPIF1<0，并求SKIPIF1<0的最小值.10．（2023秋·江蘇·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0，其中SKIPIF1<0為實數(shù)，SKIPIF1<0是自然對數(shù)的底數(shù)．(1)當SKIPIF1<0時，求曲線SKIPIF1<0在點SKIPIF1<0處的切線方程；(2)若SKIPIF1<0為SKIPIF1<0的導函數(shù)，SKIPIF1<0在SKIPIF1<0上有兩個極值點，求SKIPIF1<0的取值范圍．11．（2023·福建·統(tǒng)考一模）已知函數(shù)SKIPIF1<0．(1)討論SKIPIF1<0的極值點個數(shù)；(2)若SKIPIF1<0有兩個極值點SKIPIF1<0，且SKIPIF1<0，當SKIPIF1<0時，證明：SKIPIF1<0．12．（2023秋·安徽合肥·高三校考期末）已知函數(shù)SKIPIF1<0，SKIPIF1<0．(1)討論函數(shù)SKIPIF1<0的單調性；(2)若SKIPIF1<0，且當SKIPIF1<0時，函數(shù)SKIPIF1<0恰好有兩個極值點，求實數(shù)SKIPIF1<0的取值范圍．13．（2023春·河北邯鄲·高三校聯(lián)考開學考試）設函數(shù)SKIPIF1<0.(1)當SKIPIF1<0時，求曲線SKIPIF1<0在點SKIPIF1<0處的切線方程；(2)若SKIPIF1<0，且SKIPIF1<0在區(qū)間SKIPIF1<0上有極值，求實數(shù)a的取值范圍.14．（2023秋·江蘇南通·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0，SKIPIF1<0.(1)當SKIPIF1<0時，求SKIPIF1<0的極值；(2)當SKIPIF1<0時，設函數(shù)SKIPIF1<0的兩個極值點為SKIPIF1<0，SKIPIF1<0，證明：SKIPIF1<0.15．（2023秋·河南開封·高三統(tǒng)考期末）已知函數(shù)SKIPIF1<0．(1)討論函數(shù)SKIPIF1<0的單調性；(2)若SKIPIF1<0是函數(shù)SKIPIF1<0SKIPIF1<0的極小值點，求a的取值范圍．16．（2023秋·廣東·高三校聯(lián)考期末）已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0是SKIPIF1<0的極值點，求a；(2)若SKIPIF1<0，SKIPIF1<0分別是SKIPIF1<0的零點和極值點，當SKIPIF1<0時，證明：SKIPIF1<0．17．（2023·全國·高三專題練習）已知函數(shù)SKIPIF1<0．(1)若SKIPIF1<0是SKIPIF1<0的一個極值點，求SKIPIF1<0