第5講導數的綜合應用[考情分析]1.利用導數研究函數的單調性與極值(最值)是高考的常見題型，而導數與函數、不等式、方程、數列等的交匯命題是高考的熱點和難點.2.多以解答題的形式壓軸出現，難度較大．母題突破1導數與不等式的證明母題(2023·十堰調研)已知函數f(x)＝(2－x)ex－ax－2.(1)若f(x)在R上是減函數，求a的取值范圍；(2)當0≤a<1時，求證：f(x)在(0，＋∞)上只有一個零點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f(x)＝ex＋exlnx(其中e是自然對數的底數)．求證：f(x)≥ex2.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________[子題2]已知函數f(x)＝lnx，g(x)＝ex.證明：f(x)＋eq\f(2,ex)>g(－x)．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________規律方法利用導數證明不等式問題的方法(1)直接構造函數法：證明不等式f(x)>g(x)(或f(x)<g(x))轉化為證明f(x)－g(x)>0(或f(x)－g(x)<0)，進而構造輔助函數h(x)＝f(x)－g(x)．(2)適當放縮構造法：一是根據已知條件適當放縮；二是利用常見放縮結論．(3)構造“形似”函數，稍作變形再構造，對原不等式同結構變形，根據相似結構構造輔助函數．1．(2023·桂林模擬)已知函數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f(x)＝a(x2－1)－lnx(x>0)．若0<a<eq\f(1,2)，設函數f(x)的較大的一個零點記為x0，求證：f′(x0)<1－2a._____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________