專題一　培優(yōu)點(diǎn)4　極值點(diǎn)偏移問(wèn)題--2024年高考數(shù)學(xué)復(fù)習(xí)二輪講義-試卷下載-教習(xí)網(wǎng)

考點(diǎn)一對(duì)稱化構(gòu)造函數(shù)
例1 (2023·唐山模擬)已知函數(shù)f(x)＝xe2－x.
(1)求f(x)的極值；
(2)若a>1，b>1，a≠b，f(a)＋f(b)＝4，證明：a＋b2x0型，構(gòu)造函數(shù)F(x)＝f(x)－f(2x0－x)．
(2)對(duì)結(jié)論x1x2>xeq \\al(2,0)型，方法一是構(gòu)造函數(shù)F(x)＝f(x)－f eq \b\lc\(\rc\)(\a\vs4\al\c1(\f(x\\al(2,0),x)))，通過(guò)研究F(x)的單調(diào)性獲得不等式；方法二是兩邊取對(duì)數(shù)，轉(zhuǎn)化成ln x1＋ln x2>2ln x0，再把ln x1，ln x2看成兩變量即可．
跟蹤演練1 (2022·全國(guó)甲卷)已知函數(shù)f(x)＝eq \f(ex,x)－ln x＋x－a.
(1)若f(x)≥0，求a的取值范圍；
(2)證明：若f(x)有兩個(gè)零點(diǎn)x1，x2，則x1x22x1，求證：x1xeq \\al(2,2)>eq \f(32,e3).(參考數(shù)據(jù)：ln 2≈0.693，ln 3≈1.099)
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規(guī)律方法比值代換法是指通過(guò)代數(shù)變形將所證的雙變量不等式通過(guò)代換t＝eq \f(x1,x2)化為單變量的函數(shù)不等式，利用函數(shù)單調(diào)性證明．
跟蹤演練2 (2023·淮北模擬)已知a是實(shí)數(shù)，函數(shù)f(x)＝aln x－x.
(1)討論f(x)的單調(diào)性；
(2)若f(x)有兩個(gè)相異的零點(diǎn)x1，x2且x1>x2>0，求證：x1x2>e2.
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