2024年高考數(shù)學(xué)第一輪復(fù)習(xí)專題訓(xùn)練第三章　§3.8　隱零點(diǎn)與極值點(diǎn)偏移問題[培優(yōu)課]-試卷下載-教習(xí)網(wǎng)

§3.8　隱零點(diǎn)與極值點(diǎn)偏移問題隱零點(diǎn)問題是指對函數(shù)的零點(diǎn)設(shè)而不求，通過一種整體代換和過渡，再結(jié)合題目條件最終解決問題；極值點(diǎn)偏移是指函數(shù)在極值點(diǎn)左右的增減速度不一樣，導(dǎo)致函數(shù)圖象不具有對稱性，隱零點(diǎn)與極值點(diǎn)偏移問題常常出現(xiàn)在高考數(shù)學(xué)的壓軸題中，這類題往往對思維要求較高，過程較為煩瑣，計算量較大，難度大．題型一　隱零點(diǎn)例1　(2023·鄭州模擬)已知函數(shù)f(x)＝ex＋1－＋1，g(x)＝＋2.(1)求函數(shù)g(x)的極值；(2)當(dāng)x>0時，證明：f(x)≥g(x)．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思維升華　零點(diǎn)問題求解三步曲(1)用函數(shù)零點(diǎn)存在定理判定導(dǎo)函數(shù)零點(diǎn)的存在性，列出零點(diǎn)方程f′(x0)＝0，并結(jié)合f′(x)的單調(diào)性得到零點(diǎn)的取值范圍．(2)以零點(diǎn)為分界點(diǎn)，說明導(dǎo)函數(shù)f′(x)的正負(fù)，進(jìn)而得到f(x)的最值表達(dá)式．(3)將零點(diǎn)方程適當(dāng)變形，整體代入最值式子進(jìn)行化簡證明，有時(1)中的零點(diǎn)范圍還可以適當(dāng)縮?。?/span>跟蹤訓(xùn)練1　(2023·濰坊模擬)設(shè)函數(shù)f(x)＝x－aln x－2.(1)求f(x)的單調(diào)區(qū)間；(2)若a＝1，f′(x)為f(x)的導(dǎo)函數(shù)，當(dāng)x>1時，ln x＋1>(1＋k)f′(x)，求整數(shù)k的最大值．________________________________________________________________________________________________________________________________________________________________________________________________________________________題型二　極值點(diǎn)偏移例2　已知函數(shù)f(x)＝xe－x.(1)求函數(shù)f(x)的單調(diào)區(qū)間和極值；(2)若x1≠x2且f(x1)＝f(x2)，求證：x1＋x2>2.________________________________________________________________________________________________________________________________________________________________________________________________________________________思維升華　極值點(diǎn)偏移問題的解法(1)(對稱化構(gòu)造法)構(gòu)造輔助函數(shù)：對結(jié)論x1＋x2>(<)2x0型，構(gòu)造函數(shù)F(x)＝f(x)－f(2x0－x)；對結(jié)論x1x2>(<)x型，構(gòu)造函數(shù)F(x)＝f(x)－f ，通過研究F(x)的單調(diào)性獲得不等式．(2)(比值代換法)通過代數(shù)變形將所證的雙變量不等式通過代換t＝化為單變量的函數(shù)不等式，利用函數(shù)單調(diào)性證明．跟蹤訓(xùn)練2　已知函數(shù)f(x)＝ln(x＋a)－，函數(shù)g(x)滿足ln[g(x)＋x2]＝ln x＋x－a.(1)討論函數(shù)f(x)的單調(diào)性；(2)若g(x)有兩個不同的零點(diǎn)x1，x2，證明：x1x2<1.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________