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(shù)在點處取得極大值同理可得由一元函數(shù)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(shù)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(shù)極值的必要條件：這種說法適用于n元函數(shù)多元函數(shù)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(shù)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(shù)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(shù)但在原點(0,0)，函數(shù)沒有偏導數(shù)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(jīng)講過設證明：偏導數(shù)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(shù)無導數(shù)圓錐面在頂點無切平面原點是函數(shù)的奇點以上二元函數(shù)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(shù)極值的充分條件？回憶：一元函數(shù)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(shù)極值的充分條件定理讀書是駐點：二階偏導數(shù)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(shù)進行判定極值充分條件的證明涉及二元函數(shù)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)