2025優(yōu)化設(shè)計(jì)一輪素能培優(yōu)(六) 破解“雙變量問(wèn)題”的基本策略

素能培優(yōu)(六)破解“雙變量問(wèn)題”的基本策略高考總復(fù)習(xí)優(yōu)化設(shè)計(jì)GAOKAOZONGFUXIYOUHUASHEJI2025在近幾年的高考試題中,常常涉及“雙變量”的相關(guān)問(wèn)題,以求參數(shù)的取值范圍和證明不等式為主,這類(lèi)問(wèn)題難度較大,對(duì)能力要求較高.破解這類(lèi)問(wèn)題的關(guān)鍵:一是轉(zhuǎn)化,由已知條件入手,尋找雙變量所滿足的等量關(guān)系,將雙變量化為單變量進(jìn)行求解;二是巧妙構(gòu)造函數(shù),再借助導(dǎo)數(shù),研究函數(shù)的單調(diào)性、極值和最值,進(jìn)而解決問(wèn)題.命題點(diǎn)1轉(zhuǎn)化為函數(shù)單調(diào)性問(wèn)題求解在雙變量問(wèn)題中,如果已知函數(shù)的兩個(gè)自變量x1,x2及其對(duì)應(yīng)的函數(shù)值f(x1),f(x2)所滿足的一個(gè)不等關(guān)系,則可根據(jù)函數(shù)單調(diào)性的定義,轉(zhuǎn)化為一個(gè)與f(x)相關(guān)的函數(shù)的單調(diào)性問(wèn)題,然后利用導(dǎo)數(shù)進(jìn)行求解.例1(2024·安徽合肥模擬)設(shè)a∈R,函數(shù)f(x)=aln(-x)+(a+1)x2+1.(1)討論函數(shù)f(x)在定義域上的單調(diào)性;(2)若函數(shù)f(x)的圖象在點(diǎn)(-1,f(-1))處的切線與直線8x+y-2=0平行,且對(duì)任意x1,x2∈(-∞,0),x1≠x2,不等式

恒成立,求實(shí)數(shù)m的取值范圍.[對(duì)點(diǎn)訓(xùn)練1](2024·山東聊城模擬)已知函數(shù)f(x)=(a+1)ln

x+ax2+1.(1)當(dāng)a=2時(shí),求曲線y=f(x)在(1,f(1))處的切線方程;(2)設(shè)a≤-2,證明:對(duì)任意x1,x2∈(0,+∞),|f(x1)-f(x2)|≥4|x1-x2|.命題點(diǎn)2轉(zhuǎn)化單變量問(wèn)題求解在雙變量問(wèn)題中,如果能夠依據(jù)題目條件得出雙變量所滿足的等量關(guān)系式,則可轉(zhuǎn)化為含單變量的問(wèn)題,然后再構(gòu)造函數(shù),利用導(dǎo)數(shù)研究該函數(shù)的單調(diào)性、極值,進(jìn)而解決問(wèn)題.例2(2024·青海西寧模擬)已知函數(shù)f(x)=-x+aln

x存在兩個(gè)極值點(diǎn)x1,x2.(1)求a的取值范圍;(2)求f(x1)+f(x2)-3a的最小值.[對(duì)點(diǎn)訓(xùn)練2](2024·江蘇宿遷模擬)已知函數(shù)f(x)=x2-2ax+ln

x(a為常數(shù)).(1)當(dāng)a=1時(shí),求曲線y=f(x)在點(diǎn)(1,f(1))處的切線方程;(2)設(shè)函數(shù)f(x)的兩個(gè)極值點(diǎn)分別為x1,x2(x1<x2),求f(x2)的取值范圍.命題點(diǎn)3構(gòu)造對(duì)稱和(差)證明雙變量不等式如果雙變量問(wèn)題是證明與函數(shù)的兩個(gè)零點(diǎn)之和(之差)有關(guān)的不等式,可以利用構(gòu)造對(duì)稱和(差)的方法證明不等式,一般步驟如下:例3(2024·山西太原模擬)已知函數(shù)f(x)=ln

x++mx.(1)討論函數(shù)f(x)的單調(diào)性;(2)若f(x1)-mx1=f(x2)-mx2(0<x1<x2),求證:x1+x2>2.[對(duì)點(diǎn)訓(xùn)練3](2024·廣東佛山模擬)已知函數(shù)(1)求f(x)的單調(diào)區(qū)間和極值;(2)設(shè)f(x)的兩個(gè)零點(diǎn)為x1,x2,證明:x1+x2>2.命題點(diǎn)4構(gòu)造“根商”證明雙變量不等式[對(duì)點(diǎn)訓(xùn)練4](2024·山東日照模擬)已知函數(shù)f(x)=ln

x+.(1)求f