{VERSION 4 0 "IBM INTEL NT" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 0 1 0 
0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 0 0 0 2 0 0 
0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 
0 1 }{CSTYLE "" -1 256 "Geneva" 1 18 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 257 "Geneva" 1 12 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 258 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 
"Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Geneva" 
1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Geneva" 1 12 0 0 
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Geneva" 1 12 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 264 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 265 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 266 "Geneva" 1 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "Geneva" 
1 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "Geneva" 1 10 0 0 
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Geneva" 1 10 0 0 0 1 0 1 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 272 "Geneva" 1 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 273 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 274 "Geneva" 1 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "Geneva" 
1 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "Geneva" 1 10 0 0 
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "Geneva" 1 10 0 0 0 1 0 1 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 280 "Geneva" 1 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 281 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 282 "Geneva" 1 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "Geneva" 
1 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "Geneva" 1 10 0 0 
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "Geneva" 1 10 0 0 0 1 0 1 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 
0 0 0 1 }{CSTYLE "" -1 288 "Geneva" 1 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 289 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 290 "Geneva" 1 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 
"Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "Geneva" 
1 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "Geneva" 1 10 0 0 
0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "Geneva" 1 12 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 
1 }{CSTYLE "" -1 296 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Nor
mal" -1 0 1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 2 2 0 0 0 0 0 0 
1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 
{CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 
-1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 
"" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }
{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 
1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 
1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 
0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 
0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE 
"Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }
3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE 
"" -1 -1 "Monaco" 1 9 0 0 255 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 
0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Geneva
" 1 18 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "R3 Font 3" -1 258 1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 
1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 4
" -1 259 1 {CSTYLE "" -1 -1 "Monaco" 1 12 0 0 255 1 2 2 2 0 0 0 0 0 0 
1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 25 "\011\011\011\011\011Lagr
ange Multipliers" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 "Summary" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 257 171 "The geometry of Lagrange multipl
iers is explored in the context of the \noptimization problem for y e^
x on an ellipse. Solutions are also obtained\nnumerically using fsolve
." }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}
{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r
edefined\n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 49 "The Function f Bei
ng Optimized and the Constraint" }}{EXCHG {PARA 0 "" 0 "" {TEXT 258 
29 "The function being optimized." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "f := y*exp(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&%\"yG\"
\"\"-%$expG6#%\"xGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 27 "The cons
traint, an ellipse." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "constr1  := \+
x^2+x*y+4*y^2 -1=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(constr1G/,**
$)%\"xG\"\"#\"\"\"F+*&F)F+%\"yGF+F+*&\"\"%F+)F-F*F+F+F+!\"\"\"\"!" }}}
}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 50 "The Constraint Set With Typical \+
Level Curves of f " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "gr1 :=
 implicitplot(constr1,x=-2..2,y=-2..2,color=blue):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "display(gr1);\n" }}{PARA 13 "" 1 "" 
{GLPLOT2D 254 254 254 {PLOTDATA 2 "6$-%'CURVESG6[o7$7$$!3e0++++++))!#=
$!3!o************R\"F*7$$!3_(**********zW*F*$!3)[.++++++)!#>7$F-7$$!3C
***********\\,\"!#<$\"3gW)**********\\&F27$7$$!3KI\"R<_cp-\"F7$\"3[l**
**********zF2F47$7$F<$\"35k************zF27$$!3B++++++55F7$\"33%******
******4#F*7$7$$!36Vr&G9dG+\"F7$\"3e'************R#F*FD7$FJ7$$!3S.+++++
X%*F*$\"3/'**********\\/$F*7$7$$!3[/++++++))F*$\"3+E9dG9dGOF*FP7$7$$!3
=/++++++sF*$!3M+RC!RC!REF*7$$!3uL<_cp3EwF*$!3C.++++++CF*7$7$F\\o$!3_.+
+++++CF*F'7$FV7$$!3Ko_5Uot%f)F*$\"3sg_5Uot%z$F*7$7$$!3C1+++++!G)F*$\"3
)o*************RF*Feo7$7$$!390+++++!G)F*F^p7$$!3Fn%Q:YQ:^(F*$\"3Vf%Q:Y
Q:J%F*7$7$Fgn$\"3C8YQ:YQ:WF*Fdp7$7$Fgn$!3*3!RC!RC!REF*7$$!3%[,+++++r'F
*$!37$************)GF*7$7$$!3'Q++++++g&F*$!3!*G#p2Bp2V$F*Faq7$Fjp7$$!3
-1+++++wiF*$\"3])**********fn%F*7$7$Fhq$\"3wPC!RC!RC[F*F]r7$7$$!3c.+++
+++SF*$!334:kA'RV4%F*7$$!3Q9)eqk<TH%F*Fhr7$F\\sFgq7$Fcr7$$!3y0++++]P\\
F*$\"36**********\\P\\F*7$7$$!3+.++++++SF*$\"3]vp!zi6l/&F*Fas7$Fgr7$$!
3w)eJE0@%oQF*$!3\"*=%ot%*y:8%F*7$7$$!3!Q++++++S#F*$!3Bw6%HN#)e]%F*F]t7
$7$Fhr$\"3gwp!zi6l/&F*7$$!37!=_cp3E[$F*$\"3At@l&p3E3&F*7$7$Fbo$\"3-mmm
mmm1^F*F\\u7$7$Fdt$!3nv6%HN#)e]%F*7$$!3y3+++++b<F*$!3-)**********\\k%F
*7$7$F0$!330`Ej\"3/#[F*Fiu7$7$Fdt$\"38nmmmmm1^F*7$$!35_XXXXX&*=F*$\"3A
XXXXXX&4&F*7$7$F0$\"3G<O*[\">`D]F*Ffv7$7$F0$!3>1`Ej\"3/#[F*7$$\"3)*)[G
9dG9d\"F2$!3_bG9dG9d\\F*7$7$F>$!3<;O*[\">`D]F*Fcw7$F\\w7$$!35X'G9dG9d
\"F2$\"3udG9dG9d\\F*7$7$F>$\"3I2`Ej\"3/#[F*F]x7$Fiw7$$\"3mPXXXXX&*=F*$
!3AXXXXXX&4&F*7$7$FM$!3-mmmmmm1^F*Fgx7$7$F>$\"3u1`Ej\"3/#[F*7$$\"37#**
********\\v\"F*$\"3N,+++++XYF*7$7$FM$\"3Mx6%HN#)e]%F*Fdy7$F]y7$$\"3!o;
_cp3E[$F*$!3Ku@l&p3E3&F*7$7$F^p$!3gwp!zi6l/&F*F^z7$Fjy7$$\"3)4dJE0@%oQ
F*$\"3!GUot%*y:8%F*7$7$F^p$\"3k4:kA'RV4%F*Fhz7$7$F^p$!3sxp!zi6l/&F*7$$
\"37%*********\\P\\F*$!3y+++++]P\\F*7$7$$\"3?(************f&F*$!3)*RC!
RC!RC[F*Fe[l7$7$F\\\\l$\"3oJ#p2Bp2V$F*7$$\"3eJ)eqk<TH%F*F^p7$Fd\\lF^[l
7$7$$\"3K)************f&F*F^\\l7$$\"3#\\**********fF'F*$!3E,+++++wYF*7
$7$$\"3_(************>(F*$!3d;YQ:YQ:WF*F\\]l7$7$F\\\\l$\"3CK#p2Bp2V$F*
7$$\"3J*)**********4nF*$\"3y/+++++!*GF*7$7$Fc]l$\"37.RC!RC!REF*F[^l7$F
b]l7$$\"3Gd%Q:YQ:^(F*$!3Kj%Q:YQ:J%F*7$7$$\"3#=***********z#)F*FhrFe^l7
$F[_l7$$\"3Wf_5Uot%f)F*$!3ik_5Uot%z$F*7$7$$\"3!y************z)F*$!3MH9
dG9dGOF*F__l7$7$Ff_l$\"3!=++++++S\"F*7$$\"3tV<_cp3EwF*FM7$7$$\"3iU<_cp
3EwF*FMFa^l7$7$$\"3q'************z)F*$!3yG9dG9dGOF*7$$\"3t'**********
\\W*F*$!3q-+++++XIF*7$7$$\"3nUr&G9dG+\"F7FdtF[al7$Faal7$$\"3z*********
***45F7$!3!Q++++++5#F*7$7$$\"3bI\"R<_cp-\"F7F0Feal7$F[bl7$$\"3N++++++:
5F7$!3Y(3+++++]&F27$7$$\"31.+++++[%*F*F>F_bl7$Febl7$$\"3#*)***********
*z)F*F\\`l-%'COLOURG6&%$RGBG$\"\"!FaclF`cl$\"*++++\"!\")-%+AXESLABELSG
6$%\"xG%\"yG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 
"Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 189 "The geometric inte
rpretation of Lagrange Multipliers concerns the interplay\nbetween lev
el curves of f and the constraint set. So we begin by looking at level
 curves.\nThe level curve f = -2." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
53 "gr3 := implicitplot(f=-2,x=-2..2,y=-2..2,color=plum):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "display(\{gr1,gr3\});" }}{PARA 13 "
" 1 "" {GLPLOT2D 254 254 254 {PLOTDATA 2 "6%-%'CURVESG6Q7$7$$!3%>8$*[U
%H)>$!#?$!\"#\"\"!7$$\"3i49]o^k0B!#>$!3i9]o^k0V>!#<7$7$$\"3[l*********
***zF1$!3=sKx#pKi%=F4F.7$F67$$\"3s2\")eRSo'>)F1$!3a\")eRSo'>%=F47$7$$
\"3Gz\\dhpN7$)F1$!33++++++S=F4F<7$7$$\"3m!)\\dhpN7$)F1FE7$$\"3?vM@VX'
\\N\"!#=$!3.[8Kak\\N<F47$7$$\"3PL\\B!)=V7<FN$!3;++++++!o\"F4FK7$7$$\"3
4L\\B!)=V7<FNFU7$$\"3Y:W.%H*3c>FN$!3;UMSH*3cj\"F47$7$$\"3e'***********
*R#FN$!3g2J8AdDt:F4Fen7$7$F\\o$!3;2J8AdDt:F47$$\"3wDZlA`s)e#FN$!3FtaEK
D()Q:F47$7$$\"3c!)pb&R%3BFFN$!3C++++++?:F4Fdo7$Fjo7$$\"3HHpIa*Q_?$FN$!
3f$pIa*Q_S9F47$7$$\"3k%pY$**f&f%QFN$!3K++++++g8F4F`p7$Ffp7$$\"3d)z=I'*
*>3RFN$!3a!)=I'**>3N\"F47$7$$\"3)o*************RFN$!3Kz72#4S1M\"F4F\\q
7$Fbq7$$\"3WL'G;)*y#oXFN$!3+kG;)*y#oD\"F47$7$$\"3')e>03*R23&FN$!3Q++++
+++7F4Fhq7$7$$\"3ud>03*R23&FNFar7$$\"3DHsSRXP1`FN$!3gB2%RXP1<\"F47$7$$
\"3?(************f&FN$!3'*HwpF\"=C9\"F4Fgr7$F]s7$$\"3?CcZi&Q]/'FN$!31j
vCcQ]%3\"F47$7$$\"3-Mv3.l53lFN$!3[++++++S5F4Fcs7$7$$\"3\"H`(3.l53lFNF
\\t7$$\"34)\\$3G=FLoFN$!3K]$3G=FL+\"F47$7$$\"3_(************>(FN$!3sM%
*>>^/N(*FNFbt7$Fht7$$\"3z94zNzVRwFN$!3w@4zNzVR#*FN7$7$$\"3%=-]&zH\")z
\")FN$!3[/++++++))FNF^u7$Fdu7$$\"3e\"eq&Hi:(\\)FN$!3C)eq&Hi:(\\)FN7$7$
$\"3!y************z)FN$!3oG;jL#ecH)FNFju7$F`v7$$\"3'\\lKh#pjg$*FN$!3Ui
E8EpjgxFN7$7$$\"3>NbMv_K?5F4$!3=/++++++sFNFfv7$F\\w7$$\"31@9kKNPJ5F4$!
3!y@9kKNP6(FN7$7$$\"3\")************R5F4$!3I/c<RO4pqFNFbw7$7$Fiw$!3S0c
<RO4pqFN7$$\"3yW\\fBj\\A6F4$!3^`%\\fBj\\U'FN7$7$$\"3s*************>\"F
4$!3%Q/W#QU)Q-'FNFax7$Fgx7$$\"3[iYqPI!QA\"F4$!3IJm/x..QeFN7$7$$\"3_V\"
3d`*zp7F4$!3'Q++++++g&FNF]y7$7$$\"3uV\"3d`*zp7F4Ffy7$$\"3fuN7*>>fK\"F4
$!35_dB\"*>>f_FN7$7$$\"3m************f8F4$!3#)>62Rb@L^FNF\\z7$Fbz7$$\"
3nRFg>[]I9F4$!3w.u-'>[]q%FN7$7$$\"3d************>:F4$!39(HW!RxBuVFNFhz
7$7$F_[l$!3o(HW!RxBuVFN7$$\"3&Qku-!QGV:F4$!3%oWYF+QGB%FN7$7$$\"313IL)=
uig\"F4$!3c.++++++SFNFg[l7$7$$\"3%y+L$)=uig\"F4F`\\l7$$\"3%[ff#y[pd;F4
$!3Mcff#y[px$FN7$7$$\"3[************z;F4$!3A-#)y?&zus$FNFf\\l7$7$$\"3E
************z;F4$!3m,#)y?&zus$FN7$$\"3AV2<!='es<F4$!3&4W2<!='eK$FN7$7$
$\"3T************R=F4$!3A:%Q@_[j<$FNFg]l7$7$F^^l$!3y:%Q@_[j<$FN7$$\"3O
wfE$3ZT*=F4$!3;s(fE$3ZTHFN7$7$$\"3c**************>F4$!3;bAtkcq1FFNFf^l
-%'COLOURG6&%$RGBG$\")1Zw\"*!\")$\")PJ%y'Fg_lFe_l-F$6[o7$7$$!3e0++++++
))FN$!3!o************R\"FN7$$!3_(**********zW*FN$!3)[.++++++)F17$Fb`l7
$$!3C***********\\,\"F4$\"3gW)**********\\&F17$7$$!3KI\"R<_cp-\"F4F7Fh
`l7$7$F_al$\"35k************zF17$$!3B++++++55F4$\"33%************4#FN7
$7$$!36Vr&G9dG+\"F4F\\oFeal7$F[bl7$$!3S.+++++X%*FN$\"3/'**********\\/$
FN7$7$Fgu$\"3+E9dG9dGOFNF_bl7$7$F_w$!3M+RC!RC!REFN7$$!3uL<_cp3EwFN$!3C
.++++++CFN7$7$F]cl$!3_.++++++CFNF]`l7$Febl7$$!3Ko_5Uot%f)FN$\"3sg_5Uot
%z$FN7$7$$!3C1+++++!G)FNFcqFfcl7$7$$!390+++++!G)FNFcq7$$!3Fn%Q:YQ:^(FN
$\"3Vf%Q:YQ:J%FN7$7$F_w$\"3C8YQ:YQ:WFNFcdl7$7$F_w$!3*3!RC!RC!REFN7$$!3
%[,+++++r'FN$!37$************)GFN7$7$Ffy$!3!*G#p2Bp2V$FNF`el7$Fidl7$$!
3-1+++++wiFN$\"3])**********fn%FN7$7$Ffy$\"3wPC!RC!RC[FNFjel7$7$F`\\l$
!334:kA'RV4%FN7$$!3Q9)eqk<TH%FNF`\\l7$FgflFfel7$F`fl7$$!3y0++++]P\\FN$
\"36**********\\P\\FN7$7$$!3+.++++++SFN$\"3]vp!zi6l/&FNF\\gl7$Fdfl7$$!
3w)eJE0@%oQFN$!3\"*=%ot%*y:8%FN7$7$$!3!Q++++++S#FN$!3Bw6%HN#)e]%FNFhgl
7$7$F`\\l$\"3gwp!zi6l/&FN7$$!37!=_cp3E[$FN$\"3At@l&p3E3&FN7$7$Fccl$\"3
-mmmmmm1^FNFghl7$7$F_hl$!3nv6%HN#)e]%FN7$$!3y3+++++b<FN$!3-)**********
\\k%FN7$7$Fe`l$!330`Ej\"3/#[FNFdil7$7$F_hl$\"38nmmmmm1^FN7$$!35_XXXXX&
*=FN$\"3AXXXXXX&4&FN7$7$Fe`l$\"3G<O*[\">`D]FNFajl7$7$Fe`l$!3>1`Ej\"3/#
[FN7$$\"3)*)[G9dG9d\"F1$!3_bG9dG9d\\FN7$7$F7$!3<;O*[\">`D]FNF^[m7$Fgjl
7$$!35X'G9dG9d\"F1$\"3udG9dG9d\\FN7$7$F7$\"3I2`Ej\"3/#[FNFh[m7$Fd[m7$$
\"3mPXXXXX&*=FN$!3AXXXXXX&4&FN7$7$F\\o$!3-mmmmmm1^FNFb\\m7$7$F7$\"3u1`
Ej\"3/#[FN7$$\"37#**********\\v\"FN$\"3N,+++++XYFN7$7$F\\o$\"3Mx6%HN#)
e]%FNF_]m7$Fh\\m7$$\"3!o;_cp3E[$FN$!3Ku@l&p3E3&FN7$7$Fcq$!3gwp!zi6l/&F
NFi]m7$Fe]m7$$\"3)4dJE0@%oQFN$\"3!GUot%*y:8%FN7$7$Fcq$\"3k4:kA'RV4%FNF
c^m7$7$Fcq$!3sxp!zi6l/&FN7$$\"37%*********\\P\\FN$!3y+++++]P\\FN7$7$F^
s$!3)*RC!RC!RC[FNF`_m7$7$F^s$\"3oJ#p2Bp2V$FN7$$\"3eJ)eqk<TH%FNFcq7$F]`
mFi^m7$7$$\"3K)************f&FNFg_m7$$\"3#\\**********fF'FN$!3E,+++++w
YFN7$7$Fit$!3d;YQ:YQ:WFNFe`m7$7$F^s$\"3CK#p2Bp2V$FN7$$\"3J*)**********
4nFN$\"3y/+++++!*GFN7$7$Fit$\"37.RC!RC!REFNFbam7$F[am7$$\"3Gd%Q:YQ:^(F
N$!3Kj%Q:YQ:J%FN7$7$$\"3#=***********z#)FNF`\\lF\\bm7$Fbbm7$$\"3Wf_5Uo
t%f)FN$!3ik_5Uot%z$FN7$7$Fav$!3MH9dG9dGOFNFfbm7$7$Fav$\"3!=++++++S\"FN
7$$\"3tV<_cp3EwFNF\\o7$7$$\"3iU<_cp3EwFNF\\oFham7$7$$\"3q'************
z)FN$!3yG9dG9dGOFN7$$\"3t'**********\\W*FN$!3q-+++++XIFN7$7$$\"3nUr&G9
dG+\"F4F_hlF`dm7$Ffdm7$$\"3z************45F4$!3!Q++++++5#FN7$7$$\"3bI
\"R<_cp-\"F4Fe`lFjdm7$F`em7$$\"3N++++++:5F4$!3Y(3+++++]&F17$7$$\"31.++
+++[%*FNF7Fdem7$Fjem7$$\"3#*)************z)FNFacm-Fb_l6&Fd_l$F-F-Fcfm$
\"*++++\"Fg_l-%+AXESLABELSG6%%\"xG%\"yG%(DEFAULTG" 1 2 0 1 10 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "gr4 := implicitplot(f=-.5,x=-2..2,y
=-2..2,color=sienna):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 230 "The le
vel with value -.5 meets the ellipse obliquely. \nSo, for example at t
he point of intersection near the y axis, moving to the left along the
 ellipse  increases the value of the function while moving right decre
ases the value." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display(\{gr1,gr
3,gr4\});" }}{PARA 13 "" 1 "" {GLPLOT2D 254 254 254 {PLOTDATA 2 "6&-%'
CURVESG6Q7$7$$!3%>8$*[U%H)>$!#?$!\"#\"\"!7$$\"3i49]o^k0B!#>$!3i9]o^k0V
>!#<7$7$$\"3[l************zF1$!3=sKx#pKi%=F4F.7$F67$$\"3s2\")eRSo'>)F1
$!3a\")eRSo'>%=F47$7$$\"3Gz\\dhpN7$)F1$!33++++++S=F4F<7$7$$\"3m!)\\dhp
N7$)F1FE7$$\"3?vM@VX'\\N\"!#=$!3.[8Kak\\N<F47$7$$\"3PL\\B!)=V7<FN$!3;+
+++++!o\"F4FK7$7$$\"34L\\B!)=V7<FNFU7$$\"3Y:W.%H*3c>FN$!3;UMSH*3cj\"F4
7$7$$\"3e'************R#FN$!3g2J8AdDt:F4Fen7$7$F\\o$!3;2J8AdDt:F47$$\"
3wDZlA`s)e#FN$!3FtaEKD()Q:F47$7$$\"3c!)pb&R%3BFFN$!3C++++++?:F4Fdo7$Fj
o7$$\"3HHpIa*Q_?$FN$!3f$pIa*Q_S9F47$7$$\"3k%pY$**f&f%QFN$!3K++++++g8F4
F`p7$Ffp7$$\"3d)z=I'**>3RFN$!3a!)=I'**>3N\"F47$7$$\"3)o*************RF
N$!3Kz72#4S1M\"F4F\\q7$Fbq7$$\"3WL'G;)*y#oXFN$!3+kG;)*y#oD\"F47$7$$\"3
')e>03*R23&FN$!3Q+++++++7F4Fhq7$7$$\"3ud>03*R23&FNFar7$$\"3DHsSRXP1`FN
$!3gB2%RXP1<\"F47$7$$\"3?(************f&FN$!3'*HwpF\"=C9\"F4Fgr7$F]s7$
$\"3?CcZi&Q]/'FN$!31jvCcQ]%3\"F47$7$$\"3-Mv3.l53lFN$!3[++++++S5F4Fcs7$
7$$\"3\"H`(3.l53lFNF\\t7$$\"34)\\$3G=FLoFN$!3K]$3G=FL+\"F47$7$$\"3_(**
**********>(FN$!3sM%*>>^/N(*FNFbt7$Fht7$$\"3z94zNzVRwFN$!3w@4zNzVR#*FN
7$7$$\"3%=-]&zH\")z\")FN$!3[/++++++))FNF^u7$Fdu7$$\"3e\"eq&Hi:(\\)FN$!
3C)eq&Hi:(\\)FN7$7$$\"3!y************z)FN$!3oG;jL#ecH)FNFju7$F`v7$$\"3
'\\lKh#pjg$*FN$!3UiE8EpjgxFN7$7$$\"3>NbMv_K?5F4$!3=/++++++sFNFfv7$F\\w
7$$\"31@9kKNPJ5F4$!3!y@9kKNP6(FN7$7$$\"3\")************R5F4$!3I/c<RO4p
qFNFbw7$7$Fiw$!3S0c<RO4pqFN7$$\"3yW\\fBj\\A6F4$!3^`%\\fBj\\U'FN7$7$$\"
3s*************>\"F4$!3%Q/W#QU)Q-'FNFax7$Fgx7$$\"3[iYqPI!QA\"F4$!3IJm/
x..QeFN7$7$$\"3_V\"3d`*zp7F4$!3'Q++++++g&FNF]y7$7$$\"3uV\"3d`*zp7F4Ffy
7$$\"3fuN7*>>fK\"F4$!35_dB\"*>>f_FN7$7$$\"3m************f8F4$!3#)>62Rb
@L^FNF\\z7$Fbz7$$\"3nRFg>[]I9F4$!3w.u-'>[]q%FN7$7$$\"3d************>:F
4$!39(HW!RxBuVFNFhz7$7$F_[l$!3o(HW!RxBuVFN7$$\"3&Qku-!QGV:F4$!3%oWYF+Q
GB%FN7$7$$\"313IL)=uig\"F4$!3c.++++++SFNFg[l7$7$$\"3%y+L$)=uig\"F4F`\\
l7$$\"3%[ff#y[pd;F4$!3Mcff#y[px$FN7$7$$\"3[************z;F4$!3A-#)y?&z
us$FNFf\\l7$7$$\"3E************z;F4$!3m,#)y?&zus$FN7$$\"3AV2<!='es<F4$
!3&4W2<!='eK$FN7$7$$\"3T************R=F4$!3A:%Q@_[j<$FNFg]l7$7$F^^l$!3
y:%Q@_[j<$FN7$$\"3OwfE$3ZT*=F4$!3;s(fE$3ZTHFN7$7$$\"3c**************>F
4$!3;bAtkcq1FFNFf^l-%'COLOURG6&%$RGBG$\")1Zw\"*!\")$\")PJ%y'Fg_lFe_l-F
$6[o7$7$$!3e0++++++))FN$!3!o************R\"FN7$$!3_(**********zW*FN$!3
)[.++++++)F17$Fb`l7$$!3C***********\\,\"F4$\"3gW)**********\\&F17$7$$!
3KI\"R<_cp-\"F4F7Fh`l7$7$F_al$\"35k************zF17$$!3B++++++55F4$\"3
3%************4#FN7$7$$!36Vr&G9dG+\"F4F\\oFeal7$F[bl7$$!3S.+++++X%*FN$
\"3/'**********\\/$FN7$7$Fgu$\"3+E9dG9dGOFNF_bl7$7$F_w$!3M+RC!RC!REFN7
$$!3uL<_cp3EwFN$!3C.++++++CFN7$7$F]cl$!3_.++++++CFNF]`l7$Febl7$$!3Ko_5
Uot%f)FN$\"3sg_5Uot%z$FN7$7$$!3C1+++++!G)FNFcqFfcl7$7$$!390+++++!G)FNF
cq7$$!3Fn%Q:YQ:^(FN$\"3Vf%Q:YQ:J%FN7$7$F_w$\"3C8YQ:YQ:WFNFcdl7$7$F_w$!
3*3!RC!RC!REFN7$$!3%[,+++++r'FN$!37$************)GFN7$7$Ffy$!3!*G#p2Bp
2V$FNF`el7$Fidl7$$!3-1+++++wiFN$\"3])**********fn%FN7$7$Ffy$\"3wPC!RC!
RC[FNFjel7$7$F`\\l$!334:kA'RV4%FN7$$!3Q9)eqk<TH%FNF`\\l7$FgflFfel7$F`f
l7$$!3y0++++]P\\FN$\"36**********\\P\\FN7$7$$!3+.++++++SFN$\"3]vp!zi6l
/&FNF\\gl7$Fdfl7$$!3w)eJE0@%oQFN$!3\"*=%ot%*y:8%FN7$7$$!3!Q++++++S#FN$
!3Bw6%HN#)e]%FNFhgl7$7$F`\\l$\"3gwp!zi6l/&FN7$$!37!=_cp3E[$FN$\"3At@l&
p3E3&FN7$7$Fccl$\"3-mmmmmm1^FNFghl7$7$F_hl$!3nv6%HN#)e]%FN7$$!3y3+++++
b<FN$!3-)**********\\k%FN7$7$Fe`l$!330`Ej\"3/#[FNFdil7$7$F_hl$\"38nmmm
mm1^FN7$$!35_XXXXX&*=FN$\"3AXXXXXX&4&FN7$7$Fe`l$\"3G<O*[\">`D]FNFajl7$
7$Fe`l$!3>1`Ej\"3/#[FN7$$\"3)*)[G9dG9d\"F1$!3_bG9dG9d\\FN7$7$F7$!3<;O*
[\">`D]FNF^[m7$Fgjl7$$!35X'G9dG9d\"F1$\"3udG9dG9d\\FN7$7$F7$\"3I2`Ej\"
3/#[FNFh[m7$Fd[m7$$\"3mPXXXXX&*=FN$!3AXXXXXX&4&FN7$7$F\\o$!3-mmmmmm1^F
NFb\\m7$7$F7$\"3u1`Ej\"3/#[FN7$$\"37#**********\\v\"FN$\"3N,+++++XYFN7
$7$F\\o$\"3Mx6%HN#)e]%FNF_]m7$Fh\\m7$$\"3!o;_cp3E[$FN$!3Ku@l&p3E3&FN7$
7$Fcq$!3gwp!zi6l/&FNFi]m7$Fe]m7$$\"3)4dJE0@%oQFN$\"3!GUot%*y:8%FN7$7$F
cq$\"3k4:kA'RV4%FNFc^m7$7$Fcq$!3sxp!zi6l/&FN7$$\"37%*********\\P\\FN$!
3y+++++]P\\FN7$7$F^s$!3)*RC!RC!RC[FNF`_m7$7$F^s$\"3oJ#p2Bp2V$FN7$$\"3e
J)eqk<TH%FNFcq7$F]`mFi^m7$7$$\"3K)************f&FNFg_m7$$\"3#\\*******
***fF'FN$!3E,+++++wYFN7$7$Fit$!3d;YQ:YQ:WFNFe`m7$7$F^s$\"3CK#p2Bp2V$FN
7$$\"3J*)**********4nFN$\"3y/+++++!*GFN7$7$Fit$\"37.RC!RC!REFNFbam7$F[
am7$$\"3Gd%Q:YQ:^(FN$!3Kj%Q:YQ:J%FN7$7$$\"3#=***********z#)FNF`\\lF\\b
m7$Fbbm7$$\"3Wf_5Uot%f)FN$!3ik_5Uot%z$FN7$7$Fav$!3MH9dG9dGOFNFfbm7$7$F
av$\"3!=++++++S\"FN7$$\"3tV<_cp3EwFNF\\o7$7$$\"3iU<_cp3EwFNF\\oFham7$7
$$\"3q'************z)FN$!3yG9dG9dGOFN7$$\"3t'**********\\W*FN$!3q-++++
+XIFN7$7$$\"3nUr&G9dG+\"F4F_hlF`dm7$Ffdm7$$\"3z************45F4$!3!Q++
++++5#FN7$7$$\"3bI\"R<_cp-\"F4Fe`lFjdm7$F`em7$$\"3N++++++:5F4$!3Y(3+++
++]&F17$7$$\"31.+++++[%*FNF7Fdem7$Fjem7$$\"3#*)************z)FNFacm-Fb
_l6&Fd_l$F-F-Fcfm$\"*++++\"Fg_l-F$6ao7$7$$!3/m-[[J3)Q\"F4F+7$$!3!GWck*
4Az8F4$!3_dNa.!z2)>F47$7$Fip$!3k2w*3l'4[>F4F\\gm7$Fbgm7$$!3b0RL@\"))eK
\"F4$!3i%4m'y=6u=F47$7$$!3]1Aa)REeI\"F4FEFfgm7$7$F]hm$!3I++++++S=F47$$
!35&oubY'\\q7F4$!3Q:`UMN]p<F47$7$$!3]#)*RB([%G@\"F4FUFchm7$Fihm7$$!36,
c3iHH37F4$!3!**R9z.2<n\"F47$7$Far$!31u#o8Ye+m\"F4F]im7$Fcim7$$!3q'[t()
om.:\"F4$!3=9lA6Ljp:F47$7$$!3VCM#RGJ]6\"F4$!3-++++++?:F4Fgim7$7$F^jmF]
p7$$!3gv@k')**p'3\"F4$!37DyN8+It9F47$7$F\\t$!3N!yvr]3YT\"F4Fdjm7$Fjjm7
$$!3'=&p*G?!Q>5F4$!3s[I5(z>1Q\"F47$7$$!3D74:FL(H+\"F4FipF^[n7$7$$!3.74
:FL(H+\"F4$!35++++++g8F47$$!3ux!RPqRG`*FN$!3)G4E'Hgr'G\"F47$7$F^`l$!3a
0'3K&)\\a?\"F4F]\\n7$7$FguFd\\n7$$!3of+))))*yzx)FN$!3x%*>6,@?-7F47$7$$
!3#z1,r%47e()FNFarFh\\n7$F^]n7$$!3[Ltz?!p53)FN$!3Zn-#z4$*=6\"F47$7$$!3
Tl!3([F,LtFNF\\tFb]n7$Fh]n7$$!3!Qr(f*[,0F(FN$!3uH-/^)\\H.\"F47$7$F_w$!
3DW>K0m@F5F4F\\^n7$Fb^n7$$!3qh]uOV83lFN$!31[\\Djc'=\\*FN7$7$$!3ynQRU*
\\tl&FNFguFf^n7$F\\_n7$$!3y]A;tX+GcFN$!3edx$oU&*>x)FN7$7$Ffy$!3K30[,DO
`()FNF`_n7$Ff_n7$$!3%*p;&*e0;0[FN$!33Q$[5WR[*zFN7$7$F`\\l$!3'RN1#)[B\"
fuFNFj_n7$F``n7$$!3%e.+`?4u'QFN$!3Ws**p%z!fKtFN7$7$$!3w5]:>.8oOFNF_wFd
`n7$Fj`n7$$!3yUI:ur,cHFN$!3lkp%e#G)Rk'FN7$7$Fccl$!3)e-2;vXiN'FNF^an7$F
dan7$$!3O&Gg(RMLv>FN$!3.A(R-cmY-'FN7$7$$!3))R6Zdlra6FNFfyFhan7$7$$!3uR
6Zdlra6FNFfy7$$!3XXDr/(R-M*F1$!3%HvG&Hg(fY&FN7$7$Fe`l$!3>$zu$Q`V;aFNFe
bn7$F[cn7$$\"3SIHh\"*HzD\")F*$!3GOh\"*HzD\")[FN7$7$F7$!3W!=L>t\"e:YFNF
_cn7$Fecn7$$\"3'=6NI.%*H=\"FN$!39>^.LS*HQ%FN7$7$$\"3)4Xzgo>\">AFNF`\\l
Ficn7$F_dn7$$\"3iG;Z)=w_M#FN$!3'ejr%)=w_%RFN7$7$F\\o$!3!zwK`IRJ$RFNFcd
n7$Fidn7$$\"3%*)=P^&4IpMFN$!3g&>P^&4IpMFN7$7$Fcq$!3u(>y,B+;N$FNF]en7$F
cen7$$\"3k-&\\jh,Pm%FN$!3a4&\\jh,P1$FN7$7$F^s$!3!\\2W#>`/cGFNFgen7$F]f
n7$$\"3duF**eE2=fFN$!3h!y#**eE2=FFN7$7$$\"3i)************>(FN$!3'*e)*z
z7wLCFNFafn7$7$FitFjfn7$$\"3()pVp*3ZNA(FN$!3WwVp*3ZNU#FN7$7$$\"3hzG`6#
=(HtFNFcclF^gn7$7$FegnF_hl7$$\"3M7ggW>O'\\)FN$!3I=ggW>O'4#FN7$7$F_fm$!
3Y2zSeX\"R2#FNFign7$F_hn7$$\"3J7K2d'H3\")*FN$!3T<K2d'H3\"=FN7$7$Fiw$!3
3,Rz4MFn<FNFchn7$Fihn7$$\"3@#Gq\"4:v;6F4$!3)o#Gq\"4:vc\"FN7$7$Fhx$!3'4
,h&f5(f]\"FNF]in7$Fcin7$$\"3sUdg4w,c7F4$!3]Nu0'4w,O\"FN7$7$Fcz$!3%*zxw
%)QI$G\"FNFgin7$F]jn7$$\"3kr2(eM\\$)R\"F4$!3aAxqeM\\$=\"FN7$7$F_[l$!3G
u5wM%fN4\"FNFajn7$7$F_[l$!3Uu5wM%fN4\"FN7$$\"3())>nYW$HV:F4$!3I&*>nYW$
H.\"FN7$7$F]]l$!3a00(>!))p=$*F1F^[o7$7$F]]l$!3$p]q>!))p=$*F17$$\"3E&Gp
>lj/p\"F4$!3)HhGp>lj/*F17$7$$\"3%fB$Hl=+K=F4Fe`lF[\\o7$7$$\"3sNKHl=+K=
F4Fe`l7$$\"3bQIv5:OR=F4$!3CoRIv5:OzF17$7$F^^l$!3pOgM08(3%zF1Fh\\o7$7$F
^^l$!3ZRgM08(3%zF17$$\"3/*4'H_Jo')>F4$!3!H+5'H_JomF17$7$$\"3M*********
*****>F4$!3!zjI=;knw'F1Fe]o-Fb_l6&Fd_l$\")viobFg_l$\")%yg>%Fg_l$\")!\\
DP\"Fg_l-%+AXESLABELSG6%%\"xG%\"yG%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 56 "Setting Up and Solving the Lagran
ge Multiplier Condition" }}{EXCHG {PARA 0 "" 0 "" {TEXT 262 67 "The us
ual auxiliary function in the method of Lagrange Multipliers." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "h := f - lambda *  lhs(constr1);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,&*&%\"yG\"\"\"-%$expG6#%\"xGF(
F(*&%'lambdaGF(,**$)F,\"\"#F(F(*&F,F(F'F(F(*&\"\"%F()F'F2F(F(F(!\"\"F(
F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eqn1 := diff(h,x) = 0
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&*&%\"yG\"\"\"-%$expG6#
%\"xGF)F)*&%'lambdaGF),&F-\"\"#F(F)F)!\"\"\"\"!" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "eqn2 := diff(h,y) = 0;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eqn2G/,&-%$expG6#%\"xG\"\"\"*&%'lambdaGF+,&F*F+*&\"
\")F+%\"yGF+F+F+!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 161 "T
he optional third argument in the fsolve call allows us to only seek s
olutions in\na certain range. Our choice of range here was suggested b
y the pictures above." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "fsolve(\{e
qn1,eqn2,constr1\},\{x,y,lambda\},\{x=-1.5..1.5,y=-.5 .. .5\});" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"yG$!+yT(zE%!#5/%\"xG$\"+,j]jxF(/
%'lambdaG$!+\"R/#R#)F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 95 "These \+
statements illustrate how to conveniently extract various numbers from
 the \nsolution set." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solnset := \+
%;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%(solnsetG<%/%\"yG$!+yT(zE%!#5/%\"xG$\"+,j]jxF*/%'lambdaG$!+\"R
/#R#)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 90 "The exact order in wh
ich x,y, and lambda may vary from machine to machine or time to time.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x_val := rhs(solnset[1]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&x_valG$!+yT(zE%!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y_val := rhs(solnset[3]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&y_valG$!+\"R/#R#)!#5" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 296 104 "Substituting a set of assignments bypasses the u
ncertainty of the order in which\nx,y, and lambda appear." }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 31 "f_val :=evalf(subs(solnset,f));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%&f_valG$!+$)=bw#*!#5" }}}}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 42 "The Basic Geometry of Lagrange Multipliers" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "gr6 := implicitplot(f=f_val,
x=-2..2,y=-2..2,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "display(\{gr1,gr3,gr4,gr6\});\n" }}{PARA 13 "" 1 "" {GLPLOT2D 254 
254 254 {PLOTDATA 2 "6'-%'CURVESG6Q7$7$$!3%>8$*[U%H)>$!#?$!\"#\"\"!7$$
\"3i49]o^k0B!#>$!3i9]o^k0V>!#<7$7$$\"3[l************zF1$!3=sKx#pKi%=F4
F.7$F67$$\"3s2\")eRSo'>)F1$!3a\")eRSo'>%=F47$7$$\"3Gz\\dhpN7$)F1$!33++
++++S=F4F<7$7$$\"3m!)\\dhpN7$)F1FE7$$\"3?vM@VX'\\N\"!#=$!3.[8Kak\\N<F4
7$7$$\"3PL\\B!)=V7<FN$!3;++++++!o\"F4FK7$7$$\"34L\\B!)=V7<FNFU7$$\"3Y:
W.%H*3c>FN$!3;UMSH*3cj\"F47$7$$\"3e'************R#FN$!3g2J8AdDt:F4Fen7
$7$F\\o$!3;2J8AdDt:F47$$\"3wDZlA`s)e#FN$!3FtaEKD()Q:F47$7$$\"3c!)pb&R%
3BFFN$!3C++++++?:F4Fdo7$Fjo7$$\"3HHpIa*Q_?$FN$!3f$pIa*Q_S9F47$7$$\"3k%
pY$**f&f%QFN$!3K++++++g8F4F`p7$Ffp7$$\"3d)z=I'**>3RFN$!3a!)=I'**>3N\"F
47$7$$\"3)o*************RFN$!3Kz72#4S1M\"F4F\\q7$Fbq7$$\"3WL'G;)*y#oXF
N$!3+kG;)*y#oD\"F47$7$$\"3')e>03*R23&FN$!3Q+++++++7F4Fhq7$7$$\"3ud>03*
R23&FNFar7$$\"3DHsSRXP1`FN$!3gB2%RXP1<\"F47$7$$\"3?(************f&FN$!
3'*HwpF\"=C9\"F4Fgr7$F]s7$$\"3?CcZi&Q]/'FN$!31jvCcQ]%3\"F47$7$$\"3-Mv3
.l53lFN$!3[++++++S5F4Fcs7$7$$\"3\"H`(3.l53lFNF\\t7$$\"34)\\$3G=FLoFN$!
3K]$3G=FL+\"F47$7$$\"3_(************>(FN$!3sM%*>>^/N(*FNFbt7$Fht7$$\"3
z94zNzVRwFN$!3w@4zNzVR#*FN7$7$$\"3%=-]&zH\")z\")FN$!3[/++++++))FNF^u7$
Fdu7$$\"3e\"eq&Hi:(\\)FN$!3C)eq&Hi:(\\)FN7$7$$\"3!y************z)FN$!3
oG;jL#ecH)FNFju7$F`v7$$\"3'\\lKh#pjg$*FN$!3UiE8EpjgxFN7$7$$\"3>NbMv_K?
5F4$!3=/++++++sFNFfv7$F\\w7$$\"31@9kKNPJ5F4$!3!y@9kKNP6(FN7$7$$\"3\")*
***********R5F4$!3I/c<RO4pqFNFbw7$7$Fiw$!3S0c<RO4pqFN7$$\"3yW\\fBj\\A6
F4$!3^`%\\fBj\\U'FN7$7$$\"3s*************>\"F4$!3%Q/W#QU)Q-'FNFax7$Fgx
7$$\"3[iYqPI!QA\"F4$!3IJm/x..QeFN7$7$$\"3_V\"3d`*zp7F4$!3'Q++++++g&FNF
]y7$7$$\"3uV\"3d`*zp7F4Ffy7$$\"3fuN7*>>fK\"F4$!35_dB\"*>>f_FN7$7$$\"3m
************f8F4$!3#)>62Rb@L^FNF\\z7$Fbz7$$\"3nRFg>[]I9F4$!3w.u-'>[]q%
FN7$7$$\"3d************>:F4$!39(HW!RxBuVFNFhz7$7$F_[l$!3o(HW!RxBuVFN7$
$\"3&Qku-!QGV:F4$!3%oWYF+QGB%FN7$7$$\"313IL)=uig\"F4$!3c.++++++SFNFg[l
7$7$$\"3%y+L$)=uig\"F4F`\\l7$$\"3%[ff#y[pd;F4$!3Mcff#y[px$FN7$7$$\"3[*
***********z;F4$!3A-#)y?&zus$FNFf\\l7$7$$\"3E************z;F4$!3m,#)y?
&zus$FN7$$\"3AV2<!='es<F4$!3&4W2<!='eK$FN7$7$$\"3T************R=F4$!3A
:%Q@_[j<$FNFg]l7$7$F^^l$!3y:%Q@_[j<$FN7$$\"3OwfE$3ZT*=F4$!3;s(fE$3ZTHF
N7$7$$\"3c**************>F4$!3;bAtkcq1FFNFf^l-%'COLOURG6&%$RGBG$\")1Zw
\"*!\")$\")PJ%y'Fg_lFe_l-F$6gn7$7$$!3NIhXERl4xFNF+7$$!3s(o?CHB)[vFN$!3
mJzvqw6l>F47$7$F_w$!3=I'poi0e!>F4F``l7$Ff`l7$$!3o'4TzGQB*pFN$!3s!*e?rh
wg=F47$7$$!3g')\\>F#4-(oFNFEFj`l7$F`al7$$!3E0j]*R_hW'FN$!3%*p$\\+w%Qb<
F47$7$$!33wRHE#z0'fFNFUFdal7$Fjal7$$!3`Nw-@()zKeFN$!3+Ps*y7?nl\"F47$7$
Ffy$!3=7Q\"zU?Si\"F4F^bl7$Fdbl7$$!31tSg5pPJ_FN$!3A$fR*3B'ob\"F47$7$$!3
G*R\\)fL%*o\\FNF]pFhbl7$7$$!3Q+%\\)fL%*o\\FN$!3Y++++++?:F47$$!3w(zGY+x
Ig%FN$!3)37P&*H#pf9F47$7$F`\\l$!3u.5!3#*)*QQ\"F4Fgcl7$F]dl7$$!3LYH6<$f
(4RFN$!3/1()GoS-p8F47$7$$!3R7pp/h&z$QFNFipFadl7$7$$!3%=\"pp/h&z$QFN$!3
5++++++g8F47$$!3Wvj*=uEpD$FN$!3Aj.\"eK2VF\"F47$7$$!3G*3pPNSoe#FN$!3g++
+++++7F4F`el7$7$$!3s)3pPNSoe#FNFar7$$!30oqC>Cl0DFN$!3'RHv!eZV*=\"F47$7
$$!3_.++++++CFN$!3E=w\"*p3Gz6F4F_fl7$7$$!3C.++++++CFN$!3/=w\"*p3Gz6F47
$$!3iY&[yP\"y%z\"FN$!3.Y^@i=_+6F47$7$$!3/6)f;<W];\"FNF\\tF`gl7$7$$!3!4
\")f;<W];\"FNF\\t7$$!3#yy&p(*R'[$**F1$!38VI-g8l?5F47$7$$!3)[.++++++)F1
$!3topuoo\"\\+\"F4F]hl7$Fchl7$$!3MYe%3k^$H@F1$!31Ca\"f$[1(Q*FN7$7$$\"3
m!y%*zyg(*3&F1$!3e0++++++))FNFihl7$F_il7$$\"3IbU!Q$***)ylF1$!3IM/Q$***
)yl)FN7$7$F7$!3KI5I\"oOLc)FNFeil7$F[jl7$$\"35v.k/_g(\\\"FN$!3G$QSY?0w*
yFN7$7$F\\o$!3g+szlT>(H(FNF_jl7$Fejl7$$\"3c_+2WKt\\CFN$!3Eh+2WKt\\sFN7
$7$$\"3!y*450/[CDFNF_wFijl7$F_[m7$$\"3!onU*3@0`LFN$!33%oU*3@0`lFN7$7$F
cq$!3QfXE&oe#=iFNFc[m7$Fi[m7$$\"3)p[.Y^!=ZVFN$!3'R\\.Y^!=ZfFN7$7$$\"3r
?()4k\\1=]FNFfyF]\\m7$Fc\\m7$$\"3_6tMm^7!Q&FN$!3I>tMm^7!Q&FN7$7$F^s$!3
9#RLo^]))H&FNFg\\m7$F]]m7$$\"3,$oy+l+\"3kFN$!3O*oy+l+\"3[FN7$7$Fit$!3*
[(*zlb#Q:XFNFa]m7$Fg]m7$$\"3lpRS6Nl?vFN$!3pvRS6Nl?VFN7$7$$\"31(4N]TF\"
)Q)FN$!3+.++++++SFNF[^m7$7$$\"3=)4N]TF\")Q)FNF`\\l7$$\"3!p6([F9Rv')FN$
!3iAr[F9RvQFN7$7$Fav$!3](*fR+_vZQFNFj^m7$F`_m7$$\"3]\\37!4\\(4)*FN$!3#
\\&37!4\\(4MFN7$7$Fiw$!3w)))zapS)yKFNFd_m7$Fj_m7$$\"3RZaW'[&H,6F4$!3p!
[aW'[&H,$FN7$7$Fhx$!3USGmLP/%z#FNF^`m7$Fd`m7$$\"3f@b&H\"H[F7F4$!3!HAb&
H\"H[n#FN7$7$$\"3]tpk4-S^8F4FfflFh`m7$F^am7$$\"3)o_[y*RAe8F4$!3Iw_[y*R
AQ#FN7$7$Fcz$!3)GvSP,F4Q#FNFbam7$Fham7$$\"31HX![)QW&[\"F4$!33*HX![)QW0
#FN7$7$F_[l$!3+_5um@*)G?FNF\\bm7$7$F_[l$!3s^5um@*)G?FN7$$\"3XwC2:0^<;F
4$!3/qZs]^5v<FN7$7$F]]l$!3k'eq$ey!*G<FNFibm7$7$F]]l$!3\"peq$ey!*G<FN7$
$\"3S5/bnsq`<F4$!338T]vE2P:FN7$7$F^^l$!31p`?$4yKZ\"FNFfcm7$7$F^^l$!3Mp
`?$4yKZ\"FN7$$\"3ucs,yMU$*=F4$!3zwD<!yMUL\"FN7$7$F]_l$!3kQ\\awZWb7FNFc
dm-Fb_l6&Fd_l$\"*++++\"Fg_l$F-F-F`em-F$6[o7$7$Fbil$!3!o************R\"
FN7$$!3_(**********zW*FNFdhl7$Fgem7$$!3C***********\\,\"F4$\"3gW)*****
*****\\&F17$7$$!3KI\"R<_cp-\"F4F7F[fm7$7$Fbfm$\"35k************zF17$$!
3B++++++55F4$\"33%************4#FN7$7$$!36Vr&G9dG+\"F4F\\oFhfm7$F^gm7$
$!3S.+++++X%*FN$\"3/'**********\\/$FN7$7$Fgu$\"3+E9dG9dGOFNFbgm7$7$F_w
$!3M+RC!RC!REFN7$$!3uL<_cp3EwFNF\\gl7$7$F`hmFfflFdem7$Fhgm7$$!3Ko_5Uot
%f)FN$\"3sg_5Uot%z$FN7$7$$!3C1+++++!G)FNFcqFehm7$7$$!390+++++!G)FNFcq7
$$!3Fn%Q:YQ:^(FN$\"3Vf%Q:YQ:J%FN7$7$F_w$\"3C8YQ:YQ:WFNFbim7$7$F_w$!3*3
!RC!RC!REFN7$$!3%[,+++++r'FN$!37$************)GFN7$7$Ffy$!3!*G#p2Bp2V$
FNF_jm7$Fhim7$$!3-1+++++wiFN$\"3])**********fn%FN7$7$Ffy$\"3wPC!RC!RC[
FNFijm7$7$F`\\l$!334:kA'RV4%FN7$$!3Q9)eqk<TH%FNF`\\l7$Ff[nFejm7$F_[n7$
$!3y0++++]P\\FN$\"36**********\\P\\FN7$7$Fd^m$\"3]vp!zi6l/&FNF[\\n7$Fc
[n7$$!3w)eJE0@%oQFN$!3\"*=%ot%*y:8%FN7$7$$!3!Q++++++S#FN$!3Bw6%HN#)e]%
FNFe\\n7$7$F`\\l$\"3gwp!zi6l/&FN7$$!37!=_cp3E[$FN$\"3At@l&p3E3&FN7$7$F
ffl$\"3-mmmmmm1^FNFd]n7$7$F\\]n$!3nv6%HN#)e]%FN7$$!3y3+++++b<FN$!3-)**
********\\k%FN7$7$Fdhl$!330`Ej\"3/#[FNFa^n7$7$F\\]n$\"38nmmmmm1^FN7$$!
35_XXXXX&*=FN$\"3AXXXXXX&4&FN7$7$Fdhl$\"3G<O*[\">`D]FNF^_n7$7$Fdhl$!3>
1`Ej\"3/#[FN7$$\"3)*)[G9dG9d\"F1$!3_bG9dG9d\\FN7$7$F7$!3<;O*[\">`D]FNF
[`n7$Fd_n7$$!35X'G9dG9d\"F1$\"3udG9dG9d\\FN7$7$F7$\"3I2`Ej\"3/#[FNFe`n
7$Fa`n7$$\"3mPXXXXX&*=FN$!3AXXXXXX&4&FN7$7$F\\o$!3-mmmmmm1^FNF_an7$7$F
7$\"3u1`Ej\"3/#[FN7$$\"37#**********\\v\"FN$\"3N,+++++XYFN7$7$F\\o$\"3
Mx6%HN#)e]%FNF\\bn7$Fean7$$\"3!o;_cp3E[$FN$!3Ku@l&p3E3&FN7$7$Fcq$!3gwp
!zi6l/&FNFfbn7$Fbbn7$$\"3)4dJE0@%oQFN$\"3!GUot%*y:8%FN7$7$Fcq$\"3k4:kA
'RV4%FNF`cn7$7$Fcq$!3sxp!zi6l/&FN7$$\"37%*********\\P\\FN$!3y+++++]P\\
FN7$7$F^s$!3)*RC!RC!RC[FNF]dn7$7$F^s$\"3oJ#p2Bp2V$FN7$$\"3eJ)eqk<TH%FN
Fcq7$FjdnFfcn7$7$$\"3K)************f&FNFddn7$$\"3#\\**********fF'FN$!3
E,+++++wYFN7$7$Fit$!3d;YQ:YQ:WFNFben7$7$F^s$\"3CK#p2Bp2V$FN7$$\"3J*)**
********4nFN$\"3y/+++++!*GFN7$7$Fit$\"37.RC!RC!REFNF_fn7$Fhen7$$\"3Gd%
Q:YQ:^(FN$!3Kj%Q:YQ:J%FN7$7$$\"3#=***********z#)FNF`\\lFifn7$F_gn7$$\"
3Wf_5Uot%f)FN$!3ik_5Uot%z$FN7$7$Fav$!3MH9dG9dGOFNFcgn7$7$Fav$\"3!=++++
++S\"FN7$$\"3tV<_cp3EwFNF\\o7$7$$\"3iU<_cp3EwFNF\\oFefn7$7$$\"3q'*****
*******z)FN$!3yG9dG9dGOFN7$$\"3t'**********\\W*FN$!3q-+++++XIFN7$7$$\"
3nUr&G9dG+\"F4F\\]nF]in7$Fcin7$$\"3z************45F4$!3!Q++++++5#FN7$7
$$\"3bI\"R<_cp-\"F4FdhlFgin7$F]jn7$$\"3N++++++:5F4$!3Y(3+++++]&F17$7$$
\"31.+++++[%*FNF7Fajn7$Fgjn7$$\"3#*)************z)FNF^hn-Fb_l6&Fd_lF`e
mF`emF^em-F$6ao7$7$$!3/m-[[J3)Q\"F4F+7$$!3!GWck*4Az8F4$!3_dNa.!z2)>F47
$7$Fip$!3k2w*3l'4[>F4Ff[o7$F\\\\o7$$!3b0RL@\"))eK\"F4$!3i%4m'y=6u=F47$
7$$!3]1Aa)REeI\"F4FEF`\\o7$7$Fg\\o$!3I++++++S=F47$$!35&oubY'\\q7F4$!3Q
:`UMN]p<F47$7$$!3]#)*RB([%G@\"F4FUF]]o7$Fc]o7$$!36,c3iHH37F4$!3!**R9z.
2<n\"F47$7$Far$!31u#o8Ye+m\"F4Fg]o7$F]^o7$$!3q'[t()om.:\"F4$!3=9lA6Ljp
:F47$7$$!3VCM#RGJ]6\"F4$!3-++++++?:F4Fa^o7$7$Fh^oF]p7$$!3gv@k')**p'3\"
F4$!37DyN8+It9F47$7$F\\t$!3N!yvr]3YT\"F4F^_o7$Fd_o7$$!3'=&p*G?!Q>5F4$!
3s[I5(z>1Q\"F47$7$$!3D74:FL(H+\"F4FipFh_o7$7$$!3.74:FL(H+\"F4F^el7$$!3
ux!RPqRG`*FN$!3)G4E'Hgr'G\"F47$7$Fbil$!3a0'3K&)\\a?\"F4Fe`o7$7$FguF\\a
o7$$!3of+))))*yzx)FN$!3x%*>6,@?-7F47$7$$!3#z1,r%47e()FNFarF`ao7$Ffao7$
$!3[Ltz?!p53)FN$!3Zn-#z4$*=6\"F47$7$$!3Tl!3([F,LtFNF\\tFjao7$F`bo7$$!3
!Qr(f*[,0F(FN$!3uH-/^)\\H.\"F47$7$F_w$!3DW>K0m@F5F4Fdbo7$Fjbo7$$!3qh]u
OV83lFN$!31[\\Djc'=\\*FN7$7$$!3ynQRU*\\tl&FNFguF^co7$Fdco7$$!3y]A;tX+G
cFN$!3edx$oU&*>x)FN7$7$Ffy$!3K30[,DO`()FNFhco7$F^do7$$!3%*p;&*e0;0[FN$
!33Q$[5WR[*zFN7$7$F`\\l$!3'RN1#)[B\"fuFNFbdo7$Fhdo7$$!3%e.+`?4u'QFN$!3
Ws**p%z!fKtFN7$7$$!3w5]:>.8oOFNF_wF\\eo7$Fbeo7$$!3yUI:ur,cHFN$!3lkp%e#
G)Rk'FN7$7$Fffl$!3)e-2;vXiN'FNFfeo7$F\\fo7$$!3O&Gg(RMLv>FN$!3.A(R-cmY-
'FN7$7$$!3))R6Zdlra6FNFfyF`fo7$7$$!3uR6Zdlra6FNFfy7$$!3XXDr/(R-M*F1$!3
%HvG&Hg(fY&FN7$7$Fdhl$!3>$zu$Q`V;aFNF]go7$Fcgo7$$\"3SIHh\"*HzD\")F*$!3
GOh\"*HzD\")[FN7$7$F7$!3W!=L>t\"e:YFNFggo7$F]ho7$$\"3'=6NI.%*H=\"FN$!3
9>^.LS*HQ%FN7$7$$\"3)4Xzgo>\">AFNF`\\lFaho7$Fgho7$$\"3iG;Z)=w_M#FN$!3'
ejr%)=w_%RFN7$7$F\\o$!3!zwK`IRJ$RFNF[io7$Faio7$$\"3%*)=P^&4IpMFN$!3g&>
P^&4IpMFN7$7$Fcq$!3u(>y,B+;N$FNFeio7$F[jo7$$\"3k-&\\jh,Pm%FN$!3a4&\\jh
,P1$FN7$7$F^s$!3!\\2W#>`/cGFNF_jo7$Fejo7$$\"3duF**eE2=fFN$!3h!y#**eE2=
FFN7$7$$\"3i)************>(FN$!3'*e)*zz7wLCFNFijo7$7$FitFb[p7$$\"3()pV
p*3ZNA(FN$!3WwVp*3ZNU#FN7$7$$\"3hzG`6#=(HtFNFfflFf[p7$7$F]\\pF\\]n7$$
\"3M7ggW>O'\\)FN$!3I=ggW>O'4#FN7$7$F\\[o$!3Y2zSeX\"R2#FNFa\\p7$Fg\\p7$
$\"3J7K2d'H3\")*FN$!3T<K2d'H3\"=FN7$7$Fiw$!33,Rz4MFn<FNF[]p7$Fa]p7$$\"
3@#Gq\"4:v;6F4$!3)o#Gq\"4:vc\"FN7$7$Fhx$!3'4,h&f5(f]\"FNFe]p7$F[^p7$$
\"3sUdg4w,c7F4$!3]Nu0'4w,O\"FN7$7$Fcz$!3%*zxw%)QI$G\"FNF_^p7$Fe^p7$$\"
3kr2(eM\\$)R\"F4$!3aAxqeM\\$=\"FN7$7$F_[l$!3Gu5wM%fN4\"FNFi^p7$7$F_[l$
!3Uu5wM%fN4\"FN7$$\"3())>nYW$HV:F4$!3I&*>nYW$H.\"FN7$7$F]]l$!3a00(>!))
p=$*F1Ff_p7$7$F]]l$!3$p]q>!))p=$*F17$$\"3E&Gp>lj/p\"F4$!3)HhGp>lj/*F17
$7$$\"3%fB$Hl=+K=F4FdhlFc`p7$7$$\"3sNKHl=+K=F4Fdhl7$$\"3bQIv5:OR=F4$!3
CoRIv5:OzF17$7$F^^l$!3pOgM08(3%zF1F`ap7$7$F^^l$!3ZRgM08(3%zF17$$\"3/*4
'H_Jo')>F4$!3!H+5'H_JomF17$7$$\"3M**************>F4$!3!zjI=;knw'F1F]bp
-Fb_l6&Fd_l$\")viobFg_l$\")%yg>%Fg_l$\")!\\DP\"Fg_l-%+AXESLABELSG6%%\"
xG%\"yG%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 265 41 "At a local extremum,  the level curve of
 " }{TEXT 266 2 "f " }{TEXT 267 144 "must be tangent to the constraint
 set.\nFor if not, there would be a direction of motion along the cons
traint set for which the\ndot product with " }{TEXT 268 7 "grad f " }
{TEXT 269 22 "would be positive. So " }{TEXT 270 1 "f" }{TEXT 271 7 " \+
would " }{TEXT 272 8 "increase" }{TEXT 273 23 " in that direction and
\n" }{TEXT 274 8 "decrease" }{TEXT 275 93 " in the opposite direction.
 \n\nThis is the origin of the Lagrange multiplier condition that\n
\011\011" }{TEXT 276 22 "grad f = lambda grad g" }{TEXT 277 4 "\nat " 
}{TEXT 278 2 "x0" }{TEXT 279 4 " if " }{TEXT 280 1 "f" }{TEXT 281 35 "
  restricted to the constraint set " }{TEXT 282 12 "g = constant" }
{TEXT 283 24 " has a local extremum at" }{TEXT 284 3 " x0" }{TEXT 285 
35 ".\n\n(Technically, we also need that " }{TEXT 286 7 "grad(g)" }
{TEXT 287 14 " be nonzero at" }{TEXT 288 3 " x0" }{TEXT 289 38 ", so t
hat there is a smooth\ncurve near" }{TEXT 290 3 " x0" }{TEXT 291 51 " \+
describing a component of the constraint set near " }{TEXT 292 2 "x0" 
}{TEXT 293 2 ".)" }{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 19 "Finding the Maximum" }}{EXCHG {PARA 0 "" 0 "" {TEXT 294 
48 "Now we look for the maximum of f on the ellipse." }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 74 "solnset2 := fsolve(\{eqn1,eqn2,constr1\},\{x,y,l
ambda\},\{x=0..1.5,y=0 .. .5\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
)solnset2G<%/%\"yG$\"+Ev@9Q!#5/%\"xG$\"+'*36M[F*/%'lambdaG$\"+9)Qve%F*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "x_val := rhs(solnset2[1
]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&x_valG$\"+Ev@9Q!#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "y_val := rhs(solnset2[3]);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&y_valG$\"+9)Qve%!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f_val2 := evalf(subs(solnset2,f));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'f_val2G$\"+n87&='!#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "gr7 := implicitplot(f=f_val2
,x=-2..2,y=-2..2,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "display(\{gr1,gr7\});" }}{PARA 13 "" 1 "" {GLPLOT2D 398 398 
398 {PLOTDATA 2 "6%-%'CURVESG6N7$7$$!3[++++++S5!#<$\"3uyXtg]!*\\<F*7$$
!3'=dObA')H4\"F*$\"3T************R=F*7$F-7$$!3QAP;]'=i9\"F*$\"3I@P;]'=
i%>F*7$7$$!3%4*oQ@zJv6F*$\"37**************>F*F37$7$$!3e0++++++))!#=$
\"3#=^&yG2<\"\\\"F*7$$!3ybjW2SC0!*FB$\"3d************>:F*7$FE7$$!3hzEl
/g!=K*FB$\"3)pEl/g!=s:F*7$7$$!3F/rtE*H;+\"F*$\"3[************z;F*FK7$F
Q7$F($\"3_yXtg]!*\\<F*7$7$$!3=/++++++sFB$\"3m#zA[(=pq7F*7$$!3r8))p'*=h
5zFB$\"3U************f8F*7$7$F[o$\"3m************f8F*7$$!3[/++++++))FB
FC7$7$$!3'Q++++++g&FB$\"3Yt+#))=7G3\"F*7$$!3Cn\\8^FxcmFB$\"3s*********
****>\"F*7$F\\pFen7$7$$!3c.++++++SFB$\"3jQ$*>8o6F#*FB7$$!3T3>$p'*)R?_F
B$\"3\")************R5F*7$FhpFgo7$7$$!3_.++++++CFB$\"3M=NV#GIG'yFB7$$!
3!>R=IzLCb$FB$\"3#*)************z)FB7$7$Feq$\"3!y************z)FB7$$!3
5/++++++SFBFfp7$7$$!3)[.++++++)!#>$\"3Uj6()))>E+nFB7$$!3+&oG04(3^:FB$
\"3_(************>(FB7$Fgr7$$!3C.++++++CFBFbq7$Far7$$\"3w6D>7pIDZFdr$
\"3&Gu!y3$pu#fFB7$7$$\"3[l************zFdr$\"3lbHEQme4dFBFas7$7$$\"3e'
************R#FB$\"3%*[-O\"z)Ql[FB7$$\"3z\\REj[_/)*Fdr$\"3?(**********
**f&FB7$7$$\"3<^REj[_/)*FdrFetFgs7$F]t7$$\"3#z+nm!*)Q(p$FB$\"3'e)HL$46
EI%FB7$7$$\"3)o*************RFB$\"3;eMYR3,YTFBF\\u7$7$Fet$\"3/t`$eQ(*H
`$FB7$$\"3w2(*=BLgOVFBFcu7$F[v7$Fcu$\"3seMYR3,YTFB7$Fhu7$$\"3BH'3i!=JJ
iFB$\"3Ik8z$>)ooLFB7$7$Fjr$\"3fuMxy<i5IFBFcv7$Fiv7$$\"3db'['>*yu`)FB$
\"3QR8N!3@Dm#FB7$7$F[r$\"3#RA3(\\E[lDFBF]w7$7$F[q$\"3?T%\\c5gh=#FB7$$
\"3[!>vo[?eV*FBF^t7$FjwFcw7$Fgw7$$\"3w(*>\"HS:M1\"F*$\"3!o,!)3(f%e;#FB
7$7$F_p$\"3!y)[rbF#H'=FBF_x7$7$F_p$\"33))[rbF#H'=FB7$$\"3Ya;p$z4)e7F*$
\"3\")\\M3j?!>\"=FB7$7$Fao$\"3mgigb!yue\"FBF\\y7$7$$\"3))************f
8F*Fcy7$$\"3q\"yeR!4(*[9F*$\"39w@Tg4H5:FB7$7$FH$\"3#G.0_cfFN\"FBFiy7$F
_z7$$\"3KR7u1DnM;F*$\"3q*f(eK\\F`7FB7$7$FT$\"3_,TXhcu_6FBFcz7$Fiz7$$\"
3M(y=z'Rd;=F*$\"3(*>@\"3KgU.\"FB7$7$F0$\"31%femb]I#)*FdrF][l7$Fc[l7$$
\"3;>0.irB&*>F*$\"3!z)z%pz$Gw%)Fdr7$7$$\"3M**************>F*$\"310xb?:
lq$)FdrFg[l-%'COLOURG6&%$RGBG$\"\"!Fg\\l$\"*++++\"!\")Ff\\l-F$6[o7$7$F
@$!3!o************R\"FB7$$!3_(**********zW*FBFbr7$Fa]l7$$!3C**********
*\\,\"F*$\"3gW)**********\\&Fdr7$7$$!3KI\"R<_cp-\"F*FhsFe]l7$7$F\\^l$
\"35k************zFdr7$$!3B++++++55F*$\"33%************4#FB7$7$$!36Vr&
G9dG+\"F*F^tFb^l7$Fh^l7$$!3S.+++++X%*FB$\"3/'**********\\/$FB7$7$Fdo$
\"3+E9dG9dGOFBF\\_l7$7$Ffn$!3M+RC!RC!REFB7$$!3uL<_cp3EwFBF^s7$7$Fj_lF`
qF^]l7$Fb_l7$$!3Ko_5Uot%f)FB$\"3sg_5Uot%z$FB7$7$$!3C1+++++!G)FBFcuF_`l
7$7$$!390+++++!G)FBFcu7$$!3Fn%Q:YQ:^(FB$\"3Vf%Q:YQ:J%FB7$7$Ffn$\"3C8YQ
:YQ:WFBF\\al7$7$Ffn$!3*3!RC!RC!REFB7$$!3%[,+++++r'FB$!37$************)
GFB7$7$Fho$!3!*G#p2Bp2V$FBFial7$Fbal7$$!3-1+++++wiFB$\"3])**********fn
%FB7$7$Fho$\"3wPC!RC!RC[FBFcbl7$7$Fdp$!334:kA'RV4%FB7$$!3Q9)eqk<TH%FBF
dp7$F`clF_bl7$Fibl7$$!3y0++++]P\\FB$\"36**********\\P\\FB7$7$$!3+.++++
++SFB$\"3]vp!zi6l/&FBFecl7$F]cl7$$!3w)eJE0@%oQFB$!3\"*=%ot%*y:8%FB7$7$
$!3!Q++++++S#FB$!3Bw6%HN#)e]%FBFadl7$7$Fdp$\"3gwp!zi6l/&FB7$$!37!=_cp3
E[$FB$\"3At@l&p3E3&FB7$7$F`q$\"3-mmmmmm1^FBF`el7$7$Fhdl$!3nv6%HN#)e]%F
B7$$!3y3+++++b<FB$!3-)**********\\k%FB7$7$Fbr$!330`Ej\"3/#[FBF]fl7$7$F
hdl$\"38nmmmmm1^FB7$$!35_XXXXX&*=FB$\"3AXXXXXX&4&FB7$7$Fbr$\"3G<O*[\">
`D]FBFjfl7$7$Fbr$!3>1`Ej\"3/#[FB7$$\"3)*)[G9dG9d\"Fdr$!3_bG9dG9d\\FB7$
7$Fhs$!3<;O*[\">`D]FBFggl7$F`gl7$$!35X'G9dG9d\"Fdr$\"3udG9dG9d\\FB7$7$
Fhs$\"3I2`Ej\"3/#[FBFahl7$F]hl7$$\"3mPXXXXX&*=FB$!3AXXXXXX&4&FB7$7$F^t
$!3-mmmmmm1^FBF[il7$7$Fhs$\"3u1`Ej\"3/#[FB7$$\"37#**********\\v\"FB$\"
3N,+++++XYFB7$7$F^t$\"3Mx6%HN#)e]%FBFhil7$Fail7$$\"3!o;_cp3E[$FB$!3Ku@
l&p3E3&FB7$7$Fcu$!3gwp!zi6l/&FBFbjl7$F^jl7$$\"3)4dJE0@%oQFB$\"3!GUot%*
y:8%FB7$7$Fcu$\"3k4:kA'RV4%FBF\\[m7$7$Fcu$!3sxp!zi6l/&FB7$$\"37%******
***\\P\\FB$!3y+++++]P\\FB7$7$Fet$!3)*RC!RC!RC[FBFi[m7$7$Fet$\"3oJ#p2Bp
2V$FB7$$\"3eJ)eqk<TH%FBFcu7$Ff\\mFb[m7$7$$\"3K)************f&FBF`\\m7$
$\"3#\\**********fF'FB$!3E,+++++wYFB7$7$Fjr$!3d;YQ:YQ:WFBF^]m7$7$Fet$
\"3CK#p2Bp2V$FB7$$\"3J*)**********4nFB$\"3y/+++++!*GFB7$7$Fjr$\"37.RC!
RC!REFBF[^m7$Fd]m7$$\"3Gd%Q:YQ:^(FB$!3Kj%Q:YQ:J%FB7$7$$\"3#=**********
*z#)FBFdpFe^m7$F[_m7$$\"3Wf_5Uot%f)FB$!3ik_5Uot%z$FB7$7$F[r$!3MH9dG9dG
OFBF__m7$7$F[r$\"3!=++++++S\"FB7$$\"3tV<_cp3EwFBF^t7$7$$\"3iU<_cp3EwFB
F^tFa^m7$7$$\"3q'************z)FB$!3yG9dG9dGOFB7$$\"3t'**********\\W*F
B$!3q-+++++XIFB7$7$$\"3nUr&G9dG+\"F*FhdlFi`m7$F_am7$$\"3z************4
5F*$!3!Q++++++5#FB7$7$$\"3bI\"R<_cp-\"F*FbrFcam7$Fiam7$$\"3N++++++:5F*
$!3Y(3+++++]&Fdr7$7$$\"31.+++++[%*FBFhsF]bm7$Fcbm7$FgqFj_m-Fc\\l6&Fe\\
lFf\\lFf\\lFh\\l-%+AXESLABELSG6%%\"xG%\"yG%(DEFAULTG" 1 2 0 1 10 0 2 
9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "7 6 1" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }