高等數學微積分課件 661 多元函數的極值

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

interiorpointin

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

attainsitslocalextremumatThenThegradientthereis0ordoesnotexistGeometricinterpretationExample:FindcandidatepointsofextremepointsSolutionStationarypoint在點(1,0)處取得極小值－1看上去這是一個橢圓拋物面極小值點

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：先求駐點駐點無極值所以不是極值z=xy無極值

(0,0)是鞍點雙曲拋物面鞍點

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);證設在點處取得極大值，則一元函數在點處取得極大值同理可得由一元函數極值的必要條件所以曲面z=f(x,y)在點(x0,y0,z0)有切平面：極值的必要條件的幾何解釋則設函數z=f(x,y)在點(x0,y0)取得極值水平的切平面Geometricinterpretation多么驚人的類似！極值的必要條件的梯度形式可記為：即比較：一元函數極值的必要條件：這種說法適用于n元函數多元函數取得極值得必要條件是：梯度為零矢駐點(stationarypoint)駐點(x0,y0):推論：有偏導數的極值點必為駐點駐點就是梯度為零矢的點注1駐點不一定是極值點例如雙曲拋物面得駐點：(0,0)但z=f(0,0)=0不是極值：但在(0,0)的任何鄰域內，函數值有正有負。非極值點的駐點稱為鞍點（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);鞍點SaddlepointThesaddlepointThesaddle在鞍點處，切平面將穿過曲面注2極值點不一定是駐點例如圓錐面在原點(0,0)取得極小值因為極值點不一定有偏導數但在原點(0,0)，函數沒有偏導數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);以前曾經講過設證明：偏導數fx(0,0)和fy(0,0)不存在證不存在同理，fy(0,0)也不存在上半圓錐面無偏導數無導數圓錐面在頂點無切平面原點是函數的奇點以上二元函數的極值的概念、極值的必要條件：梯度＝零矢均可推廣到三元、四元乃至n元函數極值的充分條件？回憶：一元函數極值的充分條件極值的充分條件（二階）：是極小值是極大值二元函數極值的充分條件定理讀書是駐點：二階偏導數注：此時(x0,y0)是鞍點是極值是極小值是極大值不是極值注：此時，A與C同號可能是極值也可能不是極值即，此法不能確定f(x0,y0)是否為極值須利用更高階的偏導數進行判定極值充分條件的證明涉及二元函數的泰勒公式從略例解：先求駐點駐點有極值又所以是極小值在點(1,0)處取得極小值－1看上去這是一個橢圓拋物面極小值點

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：先求駐點駐點駐點無極值又所以是極小值有極值也是極小值同理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]);褲子？with(plots):contourplot(x^4+y^4-4*x*y,x=-1..1,y=-1..1,thickness=2,contours=50,coloring=[red,green]);鞍點例解：先求駐點駐點無極值所以不是極值z=xy無極值

(0,0)是鞍點雙曲拋物面鞍點

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);再論極值的充分條件：用Hesse矩陣設(x0,y0)是駐點:或作f(x,y)