2024學(xué)生版大二輪數(shù)學(xué)新高考提高版（京津瓊魯遼粵冀鄂湘渝閩蘇浙黑吉晉皖云豫新甘貴贛桂）專題一　第5講　母題突破1　導(dǎo)數(shù)與不等式的證明15

第5講導(dǎo)數(shù)的綜合應(yīng)用[考情分析]1.利用導(dǎo)數(shù)研究函數(shù)的單調(diào)性與極值(最值)是高考的常見題型，而導(dǎo)數(shù)與函數(shù)、不等式、方程、數(shù)列等的交匯命題是高考的熱點(diǎn)和難點(diǎn).2.多以解答題的形式壓軸出現(xiàn)，難度較大．母題突破1導(dǎo)數(shù)與不等式的證明母題(2023·十堰調(diào)研)已知函數(shù)f(x)＝(2－x)ex－ax－2.(1)若f(x)在R上是減函數(shù)，求a的取值范圍；(2)當(dāng)0≤a<1時(shí)，求證：f(x)在(0，＋∞)上只有一個(gè)零點(diǎn)x0，且x0<eq\f(e,a＋1).思路分析?f′x≤0恒成立?f′xmax≤0求解?0<x0<2?x0<\f(e,a＋1)，ax0＋x0<e?ax0＋x0<2－x0?2－x0≤e________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[子題1](2023·哈師大附中模擬)已知函數(shù)f(x)＝ex＋exlnx(其中e是自然對(duì)數(shù)的底數(shù))．求證：f(x)≥ex2.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[子題2]已知函數(shù)f(x)＝lnx，g(x)＝ex.證明：f(x)＋eq\f(2,ex)>g(－x)．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________規(guī)律方法利用導(dǎo)數(shù)證明不等式問(wèn)題的方法(1)直接構(gòu)造函數(shù)法：證明不等式f(x)>g(x)(或f(x)<g(x))轉(zhuǎn)化為證明f(x)－g(x)>0(或f(x)－g(x)<0)，進(jìn)而構(gòu)造輔助函數(shù)h(x)＝f(x)－g(x)．(2)適當(dāng)放縮構(gòu)造法：一是根據(jù)已知條件適當(dāng)放縮；二是利用常見放縮結(jié)論．(3)構(gòu)造“形似”函數(shù)，稍作變形再構(gòu)造，對(duì)原不等式同結(jié)構(gòu)變形，根據(jù)相似結(jié)構(gòu)構(gòu)造輔助函數(shù)．1．(2023·桂林模擬)已知函數(shù)f(x)＝x2－cosx，求證：f(x)＋2－eq\f(x,ex－1)>0.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________2．(2023·南昌模擬)已知函數(shù)f(x)＝a(x2－1)－lnx(x>0)．若0<a<eq\f(1,2)，設(shè)函數(shù)f(x)的較大的一個(gè)零點(diǎn)記為x0，求證：f′(x0)<1－2a._____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________