高三提高班補(bǔ)充講義

1、2014屆數(shù)學(xué)培優(yōu)班專用資料函數(shù)的凹凸性一、知識(shí)定義：定義1:設(shè)函數(shù)在區(qū)間上可導(dǎo)，若函數(shù)圖象位于每一點(diǎn)處切線的上方，則稱函)(xfD)(xf數(shù)在區(qū)間上是凹函數(shù)；若函數(shù)圖象位于每一點(diǎn)處切線的下方，則稱函數(shù))(xfD)(xf在區(qū)間上是凸函數(shù)的；)(xfD定義2:設(shè)是定義在區(qū)間上的函數(shù)若對(duì)區(qū)間上的任意兩點(diǎn)和實(shí)數(shù))(xfDD12xx(01)??總有則稱函數(shù)在區(qū)間上是凸函數(shù)1212[(1)]()(1)()fxxfxfx?????????)(xfD

2、設(shè)是定義在區(qū)間上的函數(shù)若對(duì)區(qū)間上的任意兩點(diǎn)和實(shí)數(shù)總)(xfDD12xx(01)??有則稱函數(shù)在區(qū)間上是凹函數(shù)1212[(1)]()(1)()fxxfxfx?????????)(xfD特別的:當(dāng)時(shí)12??設(shè)是定義在區(qū)間上的函數(shù)若對(duì)區(qū)間上的任意兩點(diǎn)總有)(xfDD12xx則稱函數(shù)在區(qū)間上是凸函數(shù)12121()[()()]22xxffxfx???)(xfD設(shè)是定義在區(qū)間上的函數(shù)若對(duì)區(qū)間上的任意兩點(diǎn)總有)(xfDD12xx則稱函數(shù)在區(qū)間上是凹函

3、數(shù)12121()[()()]22xxffxfx???)(xfD二、關(guān)于函數(shù)的凹凸性的充要條件設(shè)函數(shù)在區(qū)間上存在二階導(dǎo)數(shù)，則在這個(gè)區(qū)間內(nèi))(xfD)(xf1.函數(shù)是凹函數(shù)的的充要條件是.)(xfy?0)(?xf2.函數(shù)是凸函數(shù)的的充要條件是當(dāng).)(xfy?()0fx?三、函數(shù)的凹凸性有關(guān)的重要不等式—琴生(Jensen)不等式:設(shè)函數(shù)是在區(qū)間上的凸(凹)函數(shù)對(duì)于區(qū)間中的任意均有)(xfDD123nxxxx?12121()[()()()]n

4、nxxxffxfxfxnn?????????()12121()[()()()]nnxxxffxfxfxnn?????????當(dāng)且僅當(dāng)時(shí)取等號(hào).12nxxx????2014屆數(shù)學(xué)培優(yōu)班專用資料事實(shí)上故是凹函數(shù)1()ln1()0gxxgxx????()lngxxx??????122abgagbg?????????????又g(a)＋g(b)－2g()2ba?22lnln()lnlnln2ababaabbabababab?????????22

5、lnln()ln()ln22abbbabbabababab?????????題4:已知函數(shù)2()fxxaxxxRaR?????(1)若試比較的大小12(0)xx???????1212122xxfxfxf????????????與(2)若的大小與(1)一致求實(shí)數(shù)的取值范12()xx?????????1212122xxfxfxf????????????與a圍.提示(1)易證是凹函數(shù)()fx(2)當(dāng)或時(shí)易證是凹函數(shù)12(0)xx???12(0

6、)xx???()fx當(dāng)且時(shí)則或2(0)x???1(0)x???12(0)2xx????12(0)2xx????再分別代入恒成立可知????1212122xxfxfxf?????????????[0)a???題5.設(shè)f(x)是定義在區(qū)間A上的連續(xù)函數(shù)，如果對(duì)任意x1、x2∈A(x1≠x2)當(dāng)0＜＜1?時(shí)，都有f［x1(1－)x2］＞f(x1)(1－)f(x2)成立，則稱f(x)是區(qū)間A上的凸函數(shù).????(1)試判斷f(x)=－x2是否為

7、R上的凸函數(shù)，并證明你的結(jié)論；(2)設(shè)f(x)是R上的凸函數(shù)，對(duì)于任意x1、x2、x3∈R.求證：f()≥3321xxx??3)()()(321xfxfxf??(3)已知函數(shù)y=sinx在［0］上是凸函數(shù)，在△ABC中，求sinAsinBsinC的最大值.?解：任取x1≠x2.(1)f［x1(1－)x2］－f(x1)－(1－)f(x2)????=－［x1(1－)x2］2x12(1－)x22????=－2x12－(1－)2x22－2(1－