高等数学微积分课件 661 多元函数的极值

ExtremeValuesofFunctionsofSeveralVariablesReviewWhatisthegradient(gradientvector)ofafunctionz=f(x,y)atapoint(a,b)?Whatisthegeometricsignificanceofthegradientofafunctionatpoint?Howtofindthedirectionalderivativeofafunctionatapoint(a,b)?Ifz=f(x,y),FindtangentplaneatthepointIfz=f(x,y),FindtotaldifferentialatthepointRecallFunctionsofone-variableLocalMaxLocalMin.StationarypointNeither非驻点InflectionpointFunctionsoftwovariablesDomain（M0

interiorpointin

D)Localminimum,alsoglobalminimumlocalmaximum/alsoglobalmaximumLocalmaxLocalminLocalmaxLocalminLocalmaxLocalminGlobalmaxGlobalminimumAlsolocalmaximumNotlocalminiHowtoFind?NecessaryconditionRecallAnecessaryconditionisalocalextremum，TheordoesnotexistForFunctionoftwovariables?ForFunctionoftwovariablesIf

attainsitslocalextremumatThenThegradientthereis0ordoesnotexistGeometricinterpretationExample:FindcandidatepointsofextremepointsSolutionStationarypoint在点(1,0)处取得极小值－1看上去这是一个椭圆抛物面极小值点

with(plots):x_axis:=plot3d([u,0,0],u=0..3,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=0..3,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..3,v=0..0.01,thickness=2):qumian:=implicitplot3d({z=x^2-x*y+y^2-2*x+y},x=-2..3,y=-2..2,z=-2..2,scaling=constrained,style=patchcontour,numpoints=10000,contours=20):display(qumian,x_axis,y_axis,z_axis,orientation=[40,70]);contourplot(x^2-x*y+y^2-2*x+y,x=-1..3,y=-2..2,contours=30,thickness=2);SolutionStationarypointsExample:Findcandidatepointsofextremepointswith(plots):contourplot(x^4+y^4-4*x*y,x=-1..1,y=-1..1,thickness=2,contours=50,coloring=[red,green]);SaddlepointSolutionStationarypointEample:FindcandidatepointsofextremepointsCriticalpointThepartialderivativedoesnotexist,sothegradientdoesnotexistat(0,0),but…qumian:=plot3d([abs(y)*cos(t),abs(y)*sin(t),y],y=0..1,t=0..2*Pi,grid=[30,30]):x_axis:=plot3d([u,0,0],u=-1..1,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=-1..1,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..2,v=0..0.01,thickness=2):display(qumian,x_axis,y_axis,z_axis);Wenowknowhowtofindcandidatesforpossiblelocalextrema,buthowtodeterminethenatureofacandidate:whetherornotitisalocalmaximum,localminimumorneither.Asufficientcondition:secondderivativetestIfisastationarypoint：AndisalocalextremumthenLocalminimumLocalmaximumSaddlepoint,notlocalextremuminconclusiveExample:FindlocalExtremaSolution：FindcriticalpointsfirstStationarypointSothereisalocalextremumandsoLocalminimumAttainsalocalminimumat(1,0)with(plots):x_axis:=plot3d([u,0,0],u=0..3,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=0..3,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..3,v=0..0.01,thickness=2):qumian:=implicitplot3d({z=x^2-x*y+y^2-2*x+y},x=-2..3,y=-2..2,z=-2..2,scaling=constrained,style=patchcontour,numpoints=10000,contours=20):display(qumian,x_axis,y_axis,z_axis,orientation=[40,70]);contourplot(x^2-x*y+y^2-2*x+y,x=-1..3,y=-2..2,contours=30,thickness=2);StationarypointsExample:FindlocalExtremaSolution：FindcriticalpointsfirstStationarypointsnoandsoLocalminimumyesAlsolocalminimumsowith(plots):qumian:=implicitplot3d(z=x^4+y^4-4*x*y,x=-2..2,y=-2..2,z=-2..3,numpoints=5000,style=patchcontour):x_axis:=plot3d([u,0,0],u=-1..3,v=0..0.01,thickness=3):y_axis:=plot3d([0,u,0],u=-1..1.5,v=0..0.01,thickness=3):z_axis:=plot3d([0,0,u],u=-1..3,v=0..0.01,thickness=3):display(qumian,x_axis,y_axis,z_axis,orientation=[-28,45]);with(plots):contourplot(x^4+y^4-4*x*y,x=-1..1,y=-1..1,thickness=2,contours=50,coloring=[red,green]);SaddlepointFindallthelocalmaxima,localminima,andsaddlepointsofthefunctionEND例解：先求驻点驻点无极值所以不是极值z=xy无极值

(0,0)是鞍点双曲抛物面鞍点

with(plots):qumian:=implicitplot3d(x*y=z,x=-2..2,y=-2..2,z=-2..2,grid=[15,15,15],style=patchcontour,contours=20):x_axis:=plot3d([u,0,0],u=-2..3,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=-2..3,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..3,v=0..0.01,thickness=2):display(qumian,x_axis,y_axis,z_axis,orientation=[-17,66],scaling=constrained);证设在点处取得极大值，则一元函数在点处取得极大值同理可得由一元函数极值的必要条件所以曲面z=f(x,y)在点(x0,y0,z0)有切平面：极值的必要条件的几何解释则设函数z=f(x,y)在点(x0,y0)取得极值水平的切平面Geometricinterpretation多么惊人的类似！极值的必要条件的梯度形式可记为：即比较：一元函数极值的必要条件：这种说法适用于n元函数多元函数取得极值得必要条件是：梯度为零矢驻点(stationarypoint)驻点(x0,y0):推论：有偏导数的极值点必为驻点驻点就是梯度为零矢的点注1驻点不一定是极值点例如双曲抛物面得驻点：(0,0)但z=f(0,0)=0不是极值：但在(0,0)的任何邻域内，函数值有正有负。非极值点的驻点称为鞍点（saddlepoint）qumian:=implicitplot3d(x*y=z,x=-2..2,y=-2..2,z=-2..2,color=yellow,grid=[15,15,15]):pingmian:=implicitplot3d(x=0.6,x=-2..2,y=-2..2,z=-2..2,color=green):x_axis:=plot3d([u,0,0],u=-2..3,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=-2..3,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..3,v=0..0.01,thickness=2):display(qumian,x_axis,y_axis,z_axis,orientation=[-17,66],scaling=constrained);鞍点SaddlepointThesaddlepointThesaddle在鞍点处，切平面将穿过曲面注2极值点不一定是驻点例如圆锥面在原点(0,0)取得极小值因为极值点不一定有偏导数但在原点(0,0)，函数没有偏导数qumian:=plot3d([abs(y)*cos(t),abs(y)*sin(t),y],y=0..1,t=0..2*Pi,grid=[30,30]):x_axis:=plot3d([u,0,0],u=-1..1,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=-1..1,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..2,v=0..0.01,thickness=2):display(qumian,x_axis,y_axis,z_axis);以前曾经讲过设证明：偏导数fx(0,0)和fy(0,0)不存在证不存在同理，fy(0,0)也不存在上半圆锥面无偏导数无导数圆锥面在顶点无切平面原点是函数的奇点以上二元函数的极值的概念、极值的必要条件：梯度＝零矢均可推广到三元、四元乃至n元函数极值的充分条件？回忆：一元函数极值的充分条件极值的充分条件（二阶）：是极小值是极大值二元函数极值的充分条件定理读书是驻点：二阶偏导数注：此时(x0,y0)是鞍点是极值是极小值是极大值不是极值注：此时，A与C同号可能是极值也可能不是极值即，此法不能确定f(x0,y0)是否为极值须利用更高阶的偏导数进行判定极值充分条件的证明涉及二元函数的泰勒公式从略例解：先求驻点驻点有极值又所以是极小值在点(1,0)处取得极小值－1看上去这是一个椭圆抛物面极小值点

with(plots):x_axis:=plot3d([u,0,0],u=0..3,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=0..3,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..3,v=0..0.01,thickness=2):qumian:=implicitplot3d({z=x^2-x*y+y^2-2*x+y},x=-2..3,y=-2..2,z=-2..2,scaling=constrained,style=patchcontour,numpoints=10000,contours=20):display(qumian,x_axis,y_axis,z_axis,orientation=[40,70]);contourplot(x^2-x*y+y^2-2*x+y,x=-1..3,y=-2..2,contours=30,thickness=2);例解：先求驻点驻点驻点无极值又所以是极小值有极值也是极小值同理with(plots):qumian:=implicitplot3d(z=x^4+y^4-4*x*y,x=-2..2,y=-2..2,z=-2..3,numpoints=5000,style=patchcontour):x_axis:=plot3d([u,0,0],u=-1..3,v=0..0.01,thickness=3):y_axis:=plot3d([0,u,0],u=-1..1.5,v=0..0.01,thickness=3):z_axis:=plot3d([0,0,u],u=-1..3,v=0..0.01,thickness=3):display(qumian,x_axis,y_axis,z_axis,orientation=[-28,45]);裤子？with(plots):contourplot(x^4+y^4-4*x*y,x=-1..1,y=-1..1,thickness=2,contours=50,coloring=[red,green]);鞍点例解：先求驻点驻点无极值所以不是极值z=xy无极值

(0,0)是鞍点双曲抛物面鞍点

with(plots):qumian:=implicitplot3d(x*y=z,x=-2..2,y=-2..2,z=-2..2,grid=[15,15,15],style=patchcontour,contours=20):x_axis:=plot3d([u,0,0],u=-2..3,v=0..0.01,thickness=2):y_axis:=plot3d([0,u,0],u=-2..3,v=0..0.01,thickness=2):z_axis:=plot3d([0,0,u],u=0..3,v=0..0.01,thickness=2):display(qumian,x_axis,y_axis,z_axis,orientation=[-17,66],scaling=constrained);再论极值的充分条件：用Hesse矩阵设(x0,y0)是驻点:或作f(x,y)