乘数偏导数在节例中们大化了体积函数高数calc 13.pptx

13.9LagrangeMultipliers

PARTIALDERIVATIVESInExample6inSection13.8,wemaximized

avolumefunctionV=xyzsubjectto

theconstraint2xz+2yz+xy=12—whichexpressedthesideconditionthatthesurfaceareawas12m2.Inthissection,wepresentLagrange’smethodformaximizingorminimizingageneralfunctionf(x,y,z)subjecttoaconstraint

(orsidecondition)oftheformg(x,y,z)=k.

So,westartbytryingtofindtheextremevaluesoff(x,y)subjecttoaconstraintof

theformg(x,y)=k.Inotherwords,weseektheextremevalues

off(x,y)whenthepoint(x,y)isrestrictedtolie

onthelevelcurveg(x,y)=k.LAGRANGEMULTIPLIERS—TWOVARIABLES

Thefigureshowsthiscurvetogetherwithseverallevelcurvesoff.Thesehavethe

equationsf(x,y)=c,

wherec=7,8,9,

10,11Thelargestvalueofc

suchthatthelevel

curvef(x,y)=c

intersectsg(x,y)=k.LAGRANGEMULTIPLIERS—TWOVARIABLES

Itappearsthatthishappenswhenthesecurvesjusttoucheachother—thatis,whentheyhaveacommontangentline.Otherwise,thevalue

ofccouldbe

increasedfurther.LAGRANGEMULTIPLIERS—TWOVARIABLES

Thismeansthatthenormallinesat

thepoint(x0,y0)wheretheytouchareidentical.Sothegradientvectorsareparallel.Thatis,

forsomescalarλ.LAGRANGEMULTIPLIERS—TWOVARIABLES

Thiskindofargumentalsoappliestotheproblemoffindingtheextremevaluesof

f(x,y,z)subjecttotheconstraintg(x,y,z)=k.Thus,thepoint(x,y,z)isrestrictedtolieon

thelevelsurfaceSwithequationg(x,y,z)=k.LAGRANGEMULTIPLIERS—THREEVARIABLES

Supposethemaximumvalueoffis

f(x0,y0,z0)=csubjecttog(x,y,z)=k.LetCbeacurvewithvectorequation

r(t)=x(t),y(t),z(t)thatliesonthelevelsurfaceg(x,y,z)=kandpassesthrough(x0,y0,z0).Ift0istheparametervaluecorrespondingtothepoint(x0,y0,z0),then

r(t0)=x0,y0,z0LAGRANGEMULTIPLIERS—THREEVARIABLES

Thecompositefunction

h