高數(shù)極值與最值

1、oxyab)(xfy 1x2x3x4x5x6xoxyoxy0 x0 x,),()(內(nèi)有定義在設(shè)函數(shù)baxf, ),(0bax ，的一個(gè)鄰域若存在0 x0 xx , )()(0 xfxf0 x)(xf)(0 xf, )()(0 xfxf0 x)(xf)(0 xf3x1x4x2x5xOxaby41,xx52,xx3x31292)(23xxxxf1x2) 1 (f1)2(f2x12xOy1232) 1()(xxxf32)(xxf3132) 1(xx35235xx,0)( xf;521x,)( xf02xx)(xf )(xf0520033. 0)0,(),0(52),(520 x0)0(f52x33.

2、 0)(52f)(0 xf 00)()(lim0 xxxfxfxx0)(lim0 xxxfxx,0)(0知由 xf,0,00時(shí)當(dāng)xx0)(0 xxxf時(shí)，故當(dāng)00 xxx;0)( xf時(shí)，當(dāng)00 xxx,0)( xf0 x0 x0 x.)(0取極大值在xxf1) 1()(32 xxf,) 1(6)(22xxxf) 15)(1(6)(22 xxxf,0)( xf1,0, 1321xxx,06)0( f0)0(f,0) 1 () 1( ff,1)(左右鄰域內(nèi)不變號(hào)在由于xxf.1)(沒(méi)有極值在xxf1xy1O階導(dǎo)點(diǎn)有直到在若函數(shù)nxxf0)(,0)()()(0)1(00 xfxfxfn,0)(0)

3、(xfnn,0)(0)(時(shí)xfn0 x,0)(0)(時(shí)xfn0 xn0 x0 x)()()(000 xxxfxfxfnnxxnxf)(!)(00)()(0nxxo)()(0 xfxf)(0nxxonnxxnxf)(!)(00)(0 xx)(xf0 x1) 1()(32 xxf, )35(24)(2 xxxf0) 1( f1x)(xf, )sin2(212xx,20 x0 x2)0(f1xy1O maxM, )(1xf, )(2xf,(),nf x, )(af)(bf minm, )(1xf, )(2xf,(),nf x, )(af)(bf)(xf,ba)(xf,ba)1292(2 xx1224

4、)9(209681012922xx )(xxf041x250 x041x250 xxxxxf1292)(23,2541, ,)(2541Cxf)(xf, )1292(23xxx,129223xxx)(xf121862xx121862xx內(nèi)有極值可疑點(diǎn)在,)(2541xf2, 1,0321xxx,3)(321941f,0)0(f,5) 1 (f,4)2(f5)(25f0 x1x25, )2)(1(6xx, )2)(1(6xx412512xyO)()(2xfx )(x)(xf)(xxxxxf1292)(23,25412400 xCD即 )100(340052xkxky(0 x100) 由) 340

5、05(2xxky0 得 x15 2100511500|kyx ) 34005(2xxky0 得 x15 即 )100(340052xkxky(0 x100) 261bhW )(6122bdb261bhW )(6122bdb(0bd) 由0)3(6122bdW 得函數(shù)的唯一駐點(diǎn)0)3(6122bdW 得函數(shù)的唯一駐點(diǎn)db310 所以當(dāng)db31時(shí) 抗彎截面模量 W 最大 這時(shí)時(shí) 抗彎截面模量 W 最大 這時(shí)dh32 261bhW FcosFFxsin5FgFPycosF)sin5(Fg ,sincos5gF, 02sincos)()(,25. 0Pcossin)(sincos)( ,0)(arct

6、an25. 0arctan214,0)( ,)(214取最大值時(shí),sincos5gF, 02sincos)()(FPxx8 . 14 . 1arctan,8 . 1arctanx),0(x222 . 32 . 3x228 . 18 . 1x)8 . 1)(2 . 3()76. 5(4 . 122222xxx,0),0(4 . 2x4 . 18 . 1)()()(xCxRxp6123)(2xxxp得令,0)( xp586. 0221x3)(xxC,1562xx ,9)(xxRxxx6623,126)( xxp又,0)(1 xp0)(2 xpy)(xp22Ox22)24(32xx414. 3222

7、x)(xC)(xR)(xp, 0)( xp22yOx22xxxxC156)(23xxR9)()()(xCxR)(xf 0 x)(0 xf)(xf 0 x)(0 xf0)(,0)(00 xfxf)(0 xf)(0 xf0)(,0)(00 xfxf, 1)()()(lim2axafxfax)()(xfA；且0)( af)()(xfB)()(xfC)()(xfD)(xf0 x,0)0(f,2cos1)(lim0 xxfx0 x).()(xf;0)0( f)(xfy 042 yyy,0)(0 xf,0)(0 xf)(xf)(0 x,)(代入方程將xf0)(4)(00 xfxf得令,0 xx 作業(yè)作業(yè)P162 1 (5), (9); 2 ; 3 ; 5 ; 10; 14; 15 axxaxf3sin31sin)(32x)(xf0)(32 f2a )(xf2233()2sinf 時(shí)取得極大值：在2)(axf3)(32f,3coscosxxa)(3cos)cos(3232a,3sin3sin2xx 00121a上的在 1 ,0)(xf，設(shè)Nnxxnxfn,)1 ()().(limnMn)(xf,0)(