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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "restart; with(linalg
): #  2-dimensional\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the pro
tected names norm and trace have been redefined and unprotected\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Nebenbedingung:" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 25 "# g := (x,y) ->1/(x*y) ;\n" }}{PARA 0 "" 0 "" 
{TEXT -1 16 "Laplacefunktion:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "L:
=(x,y,lambda) -> x*y +lambda*(x^2+y^2-1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"LGf*6%%\"xG%\"yG%'lambdaG6\"6$%)operatorG%&arrowGF*
,&*&9$\"\"\"9%F1F1*&9&F1,(*$)F0\"\"#F1F1*$)F2F8F1F1F1!\"\"F1F1F*F*F*" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "glg1:= diff(L(x, y, lamb
da), x);\nglg2:= diff(L(x, y, lambda), y);\nglg3:= diff(L(x, y, lambda
), lambda);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%glg1G,&%\"yG\"\"\"
*(\"\"#F'%'lambdaGF'%\"xGF'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gl
g2G,&%\"xG\"\"\"*(\"\"#F'%'lambdaGF'%\"yGF'F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%glg3G,(*$)%\"xG\"\"#\"\"\"F**$)%\"yGF)F*F*F*!\"\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "L\366sen \+
des Gleichungssystems:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "lsg:=solv
e(\{glg1,glg2,glg3\}):\nallvalues(lsg[1]); \nallvalues(lsg[2]); \n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$<%/%\"xG,$*&\"\"#!\"\"F(#\"\"\"F(F)/%
\"yG,$*&F(F)F(F*F+/%'lambdaGF*<%/F%F./F-F&F0" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$<%/%\"yG,$*&\"\"#!\"\"F(#\"\"\"F(F+/%\"xGF&/%'lambdaG#F
)F(<%/F-,$*&F(F)F(F*F)/F%F3F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " g:=unapply(subs(l
sg,L(x, y,lambda)),x,y,lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"gG#!\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 22 "Hinreichende Bedingung" }}}{EXCHG {PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "H:=hessian(
L(x, y,lambda),[x,y]);\nM:= subs(lsg[1],%);\n\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"HG-%'matrixG6#7$7$,$*&\"\"#\"\"\"%'lambdaGF-F-F-7$F
-F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7$7$\"\"\"F*F
)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Min:=definite(M,positi
ve_def);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MinG%&falseG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Max:=definite(M,negative_def
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MaxG%&falseG" }}}{EXCHG 
{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "\n\nrestart; with(
linalg): #  3-dimensional\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace hav
e been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 15 "Nebenbedingung:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "g := (x,y
) ->1/(x*y) ;\n" }}{PARA 0 "" 0 "" {TEXT -1 16 "Laplacefunktion:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "L:=(x,y,z,lambda) -> x^2 + y^2 + z^
2 +lambda*(z-1/(x*y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%
\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&\"\"\"F.*&9$F.9%F.!\"\"F)F)F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGf*6&%\"xG%\"yG%\"zG%'lambdaG6\"6
$%)operatorG%&arrowGF+,**$)9$\"\"#\"\"\"F4*$)9%F3F4F4*$)9&F3F4F4*&9'F4
,&F:F4*&F4F4*&F2F4F7F4!\"\"F@F4F4F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 149 "glg1:= diff(L(x, y, z, lambda), x);\nglg2:= diff(L(x
, y, z, lambda), y);\nglg3:= diff(L(x, y, z, lambda), z);\nglg4:= diff
(L(x, y, z, lambda), lambda);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%
glg1G,&*&\"\"#\"\"\"%\"xGF(F(*(%'lambdaGF(F)!\"#%\"yG!\"\"F(" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%glg2G,&*&\"\"#\"\"\"%\"yGF(F(*(%'lambdaGF
(%\"xG!\"\"F)!\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%glg3G,&*&\"
\"#\"\"\"%\"zGF(F(%'lambdaGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gl
g4G,&%\"zG\"\"\"*&F'F'*&%\"xGF'%\"yGF'!\"\"F," }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "L\366sen des Gleichungssystems:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "lsg:=solve(\{glg1,glg2,glg3,gl
g4\}):allvalues(lsg[1]); \nallvalues(lsg[1]); \nallvalues(lsg[2]); \na
llvalues(lsg[3]); \nallvalues(lsg[4]);\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#<&/%\"yG\"\"\"/%\"xGF&/%'lambdaG!\"#/%\"zGF&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#<&/%\"yG\"\"\"/%\"xGF&/%'lambdaG!\"#/%\"zGF&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/%\"yG\"\"\"/%\"xG!\"\"/%'lambdaG\"
\"#/%\"zGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/%\"yG!\"\"/%\"xG\"\"
\"/%'lambdaG\"\"#/%\"zGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/%\"yG!
\"\"/%\"xGF&/%'lambdaG!\"#/%\"zG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " g:=una
pply(subs(lsg,L (x, y,z,lambda)),x,y,z,lambda);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gG\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "Hinreich
ende Bedingung" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "H:=hessian(L(x,y,z,lambda), [x,y,z]
);\nM := subs(lsg[1],%);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG
-%'matrixG6#7%7%,&\"\"#\"\"\"**F+F,%'lambdaGF,%\"xG!\"$%\"yG!\"\"F2,$*
(F.F,F/!\"#F1F5F2\"\"!7%F3,&F+F,**F+F,F.F,F/F2F1F0F2F67%F6F6F+" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7%7%\"\"'\"\"#\"\"!7
%F+F*F,7%F,F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Minimum:
=definite(M,positive_def);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Minim
umG%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Maximum:=defi
nite(M,negative_def);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(MaximumG%&
falseG" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
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