§3.5　利用導(dǎo)數(shù)研究恒(能)成立問(wèn)題考試要求　恒(能)成立問(wèn)題是高考的?？伎键c(diǎn)，其中不等式的恒(能)成立問(wèn)題經(jīng)常與導(dǎo)數(shù)及其幾何意義、函數(shù)、方程等相交匯，綜合考查學(xué)生分析問(wèn)題、解決問(wèn)題的能力，一般以壓軸題的形式出現(xiàn)，試題難度略大． 題型一　分離參數(shù)求參數(shù)范圍例1 (2020·全國(guó)Ⅰ)已知函數(shù)f(x)＝ex＋ax2－x.(1)當(dāng)a＝1時(shí)，討論f(x)的單調(diào)性；(2)當(dāng)x≥0時(shí)，f(x)≥x3＋1，求a的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思維升華 分離參數(shù)法解決恒(能)成立問(wèn)題的策略(1)分離變量，構(gòu)造函數(shù)，直接把問(wèn)題轉(zhuǎn)化為函數(shù)的最值問(wèn)題．(2)a≥f(x)恒成立?a≥f(x)max；a≤f(x)恒成立?a≤f(x)min；a≥f(x)能成立?a≥f(x)min；a≤f(x)能成立?a≤f(x)max.跟蹤訓(xùn)練1 (2023·蘇州質(zhì)檢)已知函數(shù)f(x)＝ax－ex(a∈R)，g(x)＝.(1)當(dāng)a＝1時(shí)，求函數(shù)f(x)的極值；(2)若存在x∈(0，＋∞)，使不等式f(x)≤g(x)－ex成立，求實(shí)數(shù)a的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________題型二　等價(jià)轉(zhuǎn)化求參數(shù)范圍例2 (2023·柳州模擬)已知函數(shù)f(x)＝ax－ln x.(1)討論函數(shù)f(x)的單調(diào)性；(2)若x＝1為函數(shù)f(x)的極值點(diǎn)，當(dāng)x∈[e，＋∞)時(shí)，不等式x[f(x)－x＋1]≤m(e－x)恒成立，求實(shí)數(shù)m的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思維升華 根據(jù)不等式恒成立構(gòu)造函數(shù)轉(zhuǎn)化成求函數(shù)的最值問(wèn)題，一般需討論參數(shù)范圍，借助函數(shù)單調(diào)性求解．跟蹤訓(xùn)練2 已知函數(shù)f(x)＝(x＋a－1)ex，g(x)＝x2＋ax，其中a為常數(shù)．若對(duì)任意的x∈[0，＋∞)，不等式f(x)≥g(x)恒成立，求實(shí)數(shù)a的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 題型三　雙變量的恒(能)成立問(wèn)題例3 (2023·石家莊質(zhì)檢)已知函數(shù)f(x)＝ax2ln x與g(x)＝x2－bx.(1)若f(x)與g(x)在x＝1處有相同的切線，求a，b，并證明f(x)≥g(x)；(2)若對(duì)?x∈[1，e]，都?b∈使f(x)≥g(x)恒成立，求a的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思維升華 “雙變量”的恒(能)成立問(wèn)題一定要正確理解其實(shí)質(zhì)，深刻挖掘內(nèi)含條件，進(jìn)行等價(jià)變換，常見(jiàn)的等價(jià)轉(zhuǎn)換有(1)?x1，x2∈D，f(x1)>g(x2)?f(x)min>g(x)max.(2)?x1∈D1，?x2∈D2，f(x1)>g(x2)?f(x)min>g(x)min.(3)?x1∈D1，?x2∈D2，f(x1)>g(x2)?f(x)max>g(x)max.跟蹤訓(xùn)練3 已知函數(shù)f(x)＝ln x－mx，g(x)＝x－(a>0)．(1)求函數(shù)f(x)的單調(diào)區(qū)間；________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2)若m＝，對(duì)?x1，x2∈[2,2e2]都有g(x1)≥f(x2)成立，求實(shí)數(shù)a的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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2024年高考數(shù)學(xué)第一輪復(fù)習(xí)專題訓(xùn)練81練第三章　§3.5　利用導(dǎo)數(shù)研究恒(能)成立問(wèn)題：

這是一份2024年高考數(shù)學(xué)第一輪復(fù)習(xí)專題訓(xùn)練81練第三章　§3.5　利用導(dǎo)數(shù)研究恒(能)成立問(wèn)題，共1頁(yè)。試卷主要包含了已知函數(shù)f＝ex.，已知函數(shù)f＝aln x－x，已知函數(shù)f＝xln x.等內(nèi)容，歡迎下載使用。

2024年高考數(shù)學(xué)第一輪復(fù)習(xí)專題訓(xùn)練第三章　§3.5　利用導(dǎo)數(shù)研究恒(能)成立問(wèn)題：

這是一份2024年高考數(shù)學(xué)第一輪復(fù)習(xí)專題訓(xùn)練第三章　§3.5　利用導(dǎo)數(shù)研究恒(能)成立問(wèn)題，共2頁(yè)。

2024年數(shù)學(xué)高考大一輪復(fù)習(xí)第三章 §3.5　利用導(dǎo)數(shù)研究恒(能)成立問(wèn)題：

這是一份2024年數(shù)學(xué)高考大一輪復(fù)習(xí)第三章 §3.5　利用導(dǎo)數(shù)研究恒(能)成立問(wèn)題，共2頁(yè)。試卷主要包含了已知函數(shù)f＝ex.，已知函數(shù)f＝aln x－x，已知函數(shù)f＝xln x.等內(nèi)容，歡迎下載使用。

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