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{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } } {SECT 0 {PARA 3 "" 0 "" {TEXT -1 63 "Critical points on the graph of a function of two variables " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Pete r Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version : 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "Critical points: local m axima and minima and saddle points " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 7 "A point" }{XPPEDIT 18 0 "``(a,b);" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 6 " is a " }{TEXT 259 14 "critical point" }{TEXT -1 17 " of the function " }{XPPEDIT 18 0 "g (x,y);" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 75 " provided that the two fi rst order partial dervatives are zero at the point" }{XPPEDIT 18 0 "`` (a,b);" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 11 ", that is, " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g[x](a,b)=0" "6#/-&%\"gG6#%\"xG6 $%\"aG%\"bG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g[y](a,b)=0" "6# /-&%\"gG6#%\"yG6$%\"aG%\"bG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "The tangent plane at a critical point is " }{TEXT 259 10 "horizontal" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "A critical point may be a " }{TEXT 259 19 "loca l maximum point" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The v alue of " }{XPPEDIT 18 0 "g(x,y);" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 76 " at a local maximum point is greater than the value at neighbourin g points. " }}{PARA 0 "" 0 "" {TEXT -1 9 "Formally," }{XPPEDIT 18 0 " \+ ``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 77 " is a local maximum point \+ if there is a sufficiently small neighbourhood U of" }{XPPEDIT 18 0 " \+ ``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "f(x, y) < f(x,y);" "6#2-%\"fG6$%\"xG%\"yG-F%6$F'F(" }{TEXT -1 5 " w hen" }{XPPEDIT 18 0 " ``(x,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 10 " is in U. 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characterize the critical points of a function " }{XPPEDIT 18 0 "phi(x,y);" "6#-%$phiG6$%\"xG%\"yG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Let" }{XPPEDIT 18 0 "`` (a,b);" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 24 " be a critical point of " } {XPPEDIT 18 0 "phi(x,y)" "6#-%$phiG6$%\"xG%\"yG" }{TEXT -1 8 " (where \+ " }{XPPEDIT 18 0 "phi[x](a,b) = 0;" "6#/-&%$phiG6#%\"xG6$%\"aG%\"bG\" \"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi[y](a,b) = 0;" "6#/-&%$phi G6#%\"yG6$%\"aG%\"bG\"\"!" }{TEXT -1 10 ") and let " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta=``" "6#/%&DeltaG%!G" } {XPPEDIT 18 0 "Delta(a,b) = phi[xx](a,b)*phi[yy](a,b)-phi[yx](a,b)*``^ 2;" "6#/-%&DeltaG6$%\"aG%\"bG,&*&-&%$phiG6#%#xxG6$F'F(\"\"\"-&F-6#%#yy G6$F'F(F1F1*&-&F-6#%#yxG6$F'F(F1*$%!G\"\"#F1!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "Delta>0" "6#2\"\"!%&DeltaG" }{TEXT -1 13 ", and either \+ " }{XPPEDIT 18 0 "phi[xx](a,b) < 0;" "6#2-&%$phiG6#%#xxG6$%\"aG%\"bG\" \"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "phi[yy](a, b) < 0;" "6#2-&%$ph iG6#%#yyG6$%\"aG%\"bG\"\"!" }{TEXT -1 6 ", then" }{XPPEDIT 18 0 "``(a, b);" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 6 " is a " }{TEXT 259 19 "local ma ximum point" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "Delta>0" "6#2\"\"!%&DeltaG" }{TEXT -1 13 ", and either \+ " }{XPPEDIT 18 0 "0 < phi[xx](a,b);" "6#2\"\"!-&%$phiG6#%#xxG6$%\"aG% \"bG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "0 < phi[yy](a,b);" "6#2\"\"!- &%$phiG6#%#yyG6$%\"aG%\"bG" }{TEXT -1 6 ", then" }{XPPEDIT 18 0 "``(a, b);" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 6 " is a " }{TEXT 259 19 "local mi nimum point" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "Delta<0" "6#2%&DeltaG\"\"!" }{TEXT -1 6 ", then" } {XPPEDIT 18 0 "``(a,b);" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 6 " is a " } {TEXT 259 12 "saddle point" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "Delta = 0;" "6#/%&DeltaG\"\"!" }{TEXT -1 66 ", no conclusion can be made about the nature of the critical point " }{XPPEDIT 18 0 " ``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 31 " withou t further investigation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Partial explanation " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 14 " Supppose that" }{XPPEDIT 18 0 " ``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 24 " is a critical point of " }{XPPEDIT 18 0 "phi(x,y);" "6#-%$phiG6$%\"xG%\"yG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 15 "Assume that if " }{TEXT 263 1 "x" }{TEXT -1 5 " and " }{TEXT 264 1 "y" }{TEXT -1 27 " are sufficiently close to " }{TEXT 269 1 "a" }{TEXT -1 5 " and " }{TEXT 270 1 "b" }{TEXT -1 24 ", then the behaviour of " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 79 " is modelled well enough by the quadratic approximating polynomial g given by: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "g(x,y) = phi(a,b)+phi[x](a,b)*( x-a)+phi[y](a,b)*(y-b)+``;" "6#/-%\"gG6$%\"xG%\"yG,*-%$phiG6$%\"aG%\"b G\"\"\"*&-&F+6#F'6$F-F.F/,&F'F/F-!\"\"F/F/*&-&F+6#F(6$F-F.F/,&F(F/F.F6 F/F/%!GF/" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[phi[xx](a,b)*(x-a)^2+2*phi[xy](a,b)*(x-a)*(y-b)+phi[yy ](a,b)*(y-b)^2];" "6#7#,(*&-&%$phiG6#%#xxG6$%\"aG%\"bG\"\"\"*$,&%\"xGF .F,!\"\"\"\"#F.F.**F3F.-&F(6#%#xyG6$F,F-F.,&F1F.F,F2F.,&%\"yGF.F-F2F.F .*&-&F(6#%#yyG6$F,F-F.*$,&F 0" "6# 2\"\"!-&%$phiG6#%#xxG6$%\"aG%\"bG" }{TEXT -1 20 " can be replaced by \+ " }{XPPEDIT 18 0 "phi[yy](a,b) > 0" "6#2\"\"!-&%$phiG6#%#yyG6$%\"aG%\" bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "Delta<0" "6#2%&DeltaG\"\"!" }{TEXT -1 76 " the signs of the square brackets in (ii) and (iii) depend on t he values of " }{TEXT 271 1 "h" }{TEXT -1 5 " and " }{TEXT 272 1 "k" } {TEXT -1 30 ". It turns out that we have a " }{TEXT 259 12 "saddle poi nt" }{TEXT -1 75 ". The approach using quadratic forms provides an exp lanation of this case. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "Consider the function f defined by " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x,y) = x^4+4*x ^2*y^2-2*x^2+2*y^2-1;" "6#/-%\"fG6$%\"xG%\"yG,,*$F'\"\"%\"\"\"*(F+F,*$ F'\"\"#F,F(F/F,*&F/F,*$F'F/F,!\"\"*&F/F,*$F(F/F,F,F,F2" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "By ca refully choosing the plotting ranges and the viewing range for a three dimensional plot of " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 94 " we can see roughly where any local maximum and minimum \+ points and saddle points are located. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "f := (x,y) -> x^4+4*x^2 *y^2-2*x^2+2*y^2-1;\nplot3d(f(x,y),x=-1.7..1.7,y=-1.15..1.15,style=pat chcontour,\n color=red,orientation=[62,53],axes=boxed,view=-2..1.5 ,\n lightmodel=light4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f Gf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),,*$)9$\"\"%\"\"\"F2*(F1F2)F0 \"\"#F2)9%F5F2F2*&F5F2F4F2!\"\"*&F5F2F6F2F2F2F9F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT3D 432 272 272 {PLOTDATA 3 "6)-%%GRIDG6&;$!#F=-%+LIGHTMODELG6#%(LIGHT_4G-%+AXESLABELSG6%%\"xG%\" yGQ!F5-%*AXESSTYLEG6#%$BOXG-%&STYLEG6#%-PATCHCONTOURG-%%VIEWG6%%(DEFAU LTGFT;$F/F>$\"#:F)-%+PROJECTIONG6%$\"#iF>$\"#`F>\"\"\"" 1 4 0 1 10 0 2 1 6 2 2 1.000000 61.000000 46.000000 1 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 63 "This plot suggests that there are three c ritical points on the " }{TEXT 287 1 "x" }{TEXT -1 60 " axis: a saddle point at the origin and two minimum points. 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7$7$$\"3k#y!G:ic]5F*FIFdbcn7$Fjbcn7$$\"39FV8H'\\bY(FTF^ccn7$7$Fe\\o$\"3;tZ>H'\\bY(FT7$$\"3a\"y!G:ic] 5F*Feq7$7$$\"3K\"y!G:ic]5F*FeqF`bcn7$Fdccn7$$\"3!\\;!GEwD=6F*$!3'f\"*4 odda4(FT7$7$$\"3&o%HH)Q0)>6F*FYFcdcn7$7$FjdcnFd`^l7$$\"3mbjCI&[S8\"F*$ !3hWnR*>FvU$FT7$7$$\"3-!eli6F*F[qFbfcn7$FhfcnFhccn-Fj]p6&F\\^pF]^p Fa^pFa^p-%+AXESLABELSG6$%\"xG%\"yG" 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" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "In order to find the exact locati on of the critical points for the function we need to determine the fi rst order partial derivatives. " }}{PARA 0 "" 0 "" {TEXT -1 45 "First \+ the partial derivative with respect to " }{TEXT 281 1 "x" }{TEXT -1 7 " . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Diff(f(x,y),x);\nvalue(%):\nfactor(%);\nf_x := unappl y(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,,*$)%\"xG\"\"% \"\"\"F+*(F*F+)F)\"\"#F+)%\"yGF.F+F+*&F.F+F-F+!\"\"*&F.F+F/F+F+F+F2F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"xG\"\"\",(*$)F%\"\"#F&F&*&F *F&)%\"yGF*F&F&F&!\"\"F&\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f_ xGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&9$\"\"\",(*$)F/\"\"#F0F0 *&F4F0)9%F4F0F0F0!\"\"F0\"\"%F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 " . . . and then the partial derivative \+ with respect to " }{TEXT 282 1 "y" }{TEXT -1 7 " . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Diff(f( x,y),y);\nvalue(%):\nfactor(%);\nf_y := unapply(%,x,y);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%%DiffG6$,,*$)%\"xG\"\"%\"\"\"F+*(F*F+)F)\"\"#F +)%\"yGF.F+F+*&F.F+F-F+!\"\"*&F.F+F/F+F+F+F2F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"yG\"\"\",&*$)%\"xG\"\"#F&F+F&F&F&\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f_yGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrow GF),$*&9%\"\"\",&*$)9$\"\"#F0F5F0F0F0\"\"%F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The critical points are t he points for which f" }{XPPEDIT 18 0 "``[x](x,y)" "6#-&%!G6#%\"xG6$F' %\"yG" }{TEXT -1 6 " and f" }{XPPEDIT 18 0 "``[y](x,y)" "6#-&%!G6#%\"y G6$%\"xGF'" }{TEXT -1 15 " are both zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "solve(\{f_x(x,y)=0,f_ y(x,y)=0\},\{x,y\}):\nsols := map(allvalues,[%]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solsG7)<$/%\"xG\"\"!/%\"yGF)<$/F(\"\"\"F*<$/F(!\"\"F *<$/F(*&^##F.\"\"#F.-%%sqrtG6#F7F./F+,$*$-F96#\"\"$F.F6<$F;/F(*&^##F1F 7F.F8F.<$F3/F+,$F=FE<$FGFB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 " Thus we see that there are three critical points:" }{XPPEDIT 18 0 "``( 0,0),``(1,0);" "6$-%!G6$\"\"!F&-F$6$\"\"\"F&" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "``(-1,0);" "6#-%!G6$,$\"\"\"!\"\"\"\"!" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Inspe ction of the plots made above suggest that the points" }{XPPEDIT 18 0 "``(1,0);" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "`` (-1,0);" "6#-%!G6$,$\"\"\"!\"\"\"\"!" }{TEXT -1 28 " are minimum point s and that" }{XPPEDIT 18 0 " ``(0,0) " "6#-%!G6$\"\"!F&" }{TEXT -1 20 " is a saddle point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 81 "We can apply the following standard criteria to ch aracterize the critical points." }}{PARA 0 "" 0 "" {TEXT -1 2 "If" } {XPPEDIT 18 0 " ``(a,b) " "6#-%!G6$%\"aG%\"bG" }{TEXT -1 25 " is a cri tical point of " }{XPPEDIT 18 0 "phi(x,y);" "6#-%$phiG6$%\"xG%\"yG" } {TEXT -1 5 " and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Delta=``" "6#/%&DeltaG%!G" }{XPPEDIT 18 0 "Delta(a,b) = ``;" "6#/-%&De ltaG6$%\"aG%\"bG%!G" }{XPPEDIT 18 0 "phi[xx](a,b)*phi[yy](a,b)-phi[xy] (a,b)*``^2;" "6#,&*&-&%$phiG6#%#xxG6$%\"aG%\"bG\"\"\"-&F'6#%#yyG6$F+F, F-F-*&-&F'6#%#xyG6$F+F,F-*$%!G\"\"#F-!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 5 "then:" }}{PARA 15 "" 0 "" {TEXT -1 5 " if " } {XPPEDIT 18 0 "0 < Delta;" "6#2\"\"!%&DeltaG" }{TEXT -1 8 " and " } {XPPEDIT 18 0 "0 < phi[xx](a,b);" "6#2\"\"!-&%$phiG6#%#xxG6$%\"aG%\"bG " }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 7 " is a " }{TEXT 259 13 "minimum point" }{TEXT -1 2 ", " } }{PARA 15 "" 0 "" {TEXT -1 5 " if " }{XPPEDIT 18 0 "0 < Delta;" "6#2 \"\"!%&DeltaG" }{TEXT -1 11 " > 0 and " }{XPPEDIT 18 0 "phi[xx](a,b) < 0;" "6#2-&%$phiG6#%#xxG6$%\"aG%\"bG\"\"!" }{TEXT -1 7 ", then " } {XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 7 " is a " } {TEXT 259 13 "maximum point" }{TEXT -1 2 ", " }}{PARA 15 "" 0 "" {TEXT -1 5 " if " }{XPPEDIT 18 0 "Delta < 0;" "6#2%&DeltaG\"\"!" } {TEXT -1 8 ", then " }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" } {TEXT -1 7 " is a " }{TEXT 259 12 "saddle point" }{TEXT -1 4 ", " } }{PARA 15 "" 0 "" {TEXT -1 5 " if " }{XPPEDIT 18 0 "Delta = 0;" "6#/% &DeltaG\"\"!" }{TEXT -1 66 ", no conclusion can be made about the natu re of the critical point" }{XPPEDIT 18 0 " ``(a,b)" "6#-%!G6$%\"aG%\"b G" }{TEXT -1 31 " without further investigation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "f := (x,y) - > x^4+4*x^2*y^2-2*x^2+2*y^2-1:\nDiff(f(x,y),x$2);\nvalue(%):\nf_xx := \+ unapply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,,*$)%\"x G\"\"%\"\"\"F+*(F*F+)F)\"\"#F+)%\"yGF.F+F+*&F.F+F-F+!\"\"*&F.F+F/F+F+F +F2-%\"$G6$F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%f_xxGf*6$%\"xG% \"yG6\"6$%)operatorG%&arrowGF),(*$)9$\"\"#\"\"\"\"#7*&\"\")F2)9%F1F2F2 \"\"%!\"\"F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Diff(f(x,y),x,y);\nvalue(%):\nf_xy := unapply (%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6%,,*$)%\"xG\"\"% \"\"\"F+*(F*F+)F)\"\"#F+)%\"yGF.F+F+*&F.F+F-F+!\"\"*&F.F+F/F+F+F+F2F)F 0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%f_xyGf*6$%\"xG%\"yG6\"6$%)oper atorG%&arrowGF),$*&9%\"\"\"9$F0\"#;F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Diff(f(x,y),y$2); \nvalue(%):\nf_yy := unapply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%%DiffG6$,,*$)%\"xG\"\"%\"\"\"F+*(F*F+)F)\"\"#F+)%\"yGF.F+F+*&F.F+F -F+!\"\"*&F.F+F/F+F+F+F2-%\"$G6$F0F." }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%f_yyGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*$)9$\"\"#\"\"\"\" \")\"\"%F2F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f_xx(x,y)*f_yy(x,y)-f_xy(x,y)^2;\nDelta := un apply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,(*$)%\"xG\"\"#\" \"\"\"#7*&\"\")F*)%\"yGF)F*F*\"\"%!\"\"F*,&F&F-F0F*F*F**(\"$c#F*F'F*F. F*F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaGf*6$%\"xG%\"yG6\"6$%) operatorG%&arrowGF),&*&,(*$)9$\"\"#\"\"\"\"#7*&\"\")F4)9%F3F4F4\"\"%! \"\"F4,&F0F7F:F4F4F4*(\"$c#F4F1F4F8F4F;F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We first apply the test f or the critical point" }{XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\"\"!F&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Delta(0,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#! #;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thu s " }{XPPEDIT 18 0 "Delta < 0;" "6#2%&DeltaG\"\"!" }{TEXT -1 13 " at t he point" }{XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 21 ", so this point is a " }{TEXT 286 6 "saddle" }{TEXT -1 8 " point. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Now we ch eck the values of f" }{XPPEDIT 18 0 "``[xx];" "6#&%!G6#%#xxG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "Delta" "6#%&DeltaG" }{TEXT -1 14 " at \+ the point" }{XPPEDIT 18 0 "``(1,0);" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "[f_xx(1,0),simplify(Delta(1,0))];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7$\"\")\"#'*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "0<``" "6#2\"\"!%!G " }{TEXT -1 1 "f" }{XPPEDIT 18 0 "``[xx](1,0);" "6#-&%!G6#%#xxG6$\"\" \"\"\"!" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "0 < Delta(1,0);" "6#2\" \"!-%&DeltaG6$\"\"\"F$" }{TEXT -1 13 ", we see that" }{XPPEDIT 18 0 "` `(1,0);" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 6 " is a " }{TEXT 285 7 "mini mum" }{TEXT -1 8 " point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Next we check the critical point" }{XPPEDIT 18 0 "``(-1,0);" "6#-%!G6$,$\"\"\"!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "[f_xx (-1,0),simplify(Delta(-1,0))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\" \")\"#'*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }{XPPEDIT 18 0 "``(-1,0);" "6#-%!G6$,$\"\"\"!\"\"\"\"!" } {TEXT -1 11 " is also a " }{TEXT 284 7 "minimum" }{TEXT -1 8 " point. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "We ca n collect all the required information regarding the sign of f" } {XPPEDIT 18 0 "``[x,y]" "6#&%!G6$%\"xG%\"yG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "Delta" "6#%&DeltaG" }{TEXT -1 26 ", together with value s of " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 57 " a t the three critical points using the Maple procedures " }{TEXT 0 6 "s ignum" }{TEXT -1 5 " and " }{TEXT 0 3 "zip" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "x vals := [0,1,-1]:\nyvals := [0,0,0]:\nzip((x,y)->``(x,y),xvals,yvals); \nzip(signum@f_xx,xvals,yvals);\nzip(signum@Delta,xvals,yvals);\nzip(f ,xvals,yvals);\nevalf(%,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%!G6 $\"\"!F'-F%6$\"\"\"F'-F%6$!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 %!\"\"\"\"\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%!\"\"\"\"\"F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%!\"\"!\"#F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$!\"\"\"\"!$!\"#F&F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 47 "The critical points are classified as follows: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 9 " \+ f" }{XPPEDIT 18 0 "``[xx](a,b);" "6#-&%!G6#%#xxG6$%\"aG%\"bG" } {TEXT -1 9 " " }{XPPEDIT 18 0 "Delta" "6#%&DeltaG" }{TEXT -1 15 " " }{TEXT 283 10 "Conclusion" }{TEXT -1 22 " \+ " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }} {PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(0,0)" "6#-%!G6$\" \"!F&" }{TEXT -1 30 " 0 " }{XPPEDIT 18 0 "` `-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 48 " Saddle Point \+ 0" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(1, 0);" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 8 " " } {XPPEDIT 18 0 "``+``" "6#,&%!G\"\"\"F$F%" }{TEXT -1 17 " \+ " }{XPPEDIT 18 0 "``+``" "6#,&%!G\"\"\"F$F%" }{TEXT -1 48 " \+ Minimum -2" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(-1, 0);" "6#-%!G6$,$\"\"\"!\"\"\"\"!" } {TEXT -1 7 " " }{XPPEDIT 18 0 "``+``" "6#,&%!G\"\"\"F$F%" } {TEXT -1 16 " " }{XPPEDIT 18 0 "``+``" "6#,&%!G\"\"\"F$ F%" }{TEXT -1 48 " Minimum -2" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 " " 0 "" {TEXT -1 128 "Acknowlegement: The example in this section is ad apted from an example in a Maple worksheet from the University of N. C arolina. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Consider the function f defined by" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x,y) = x y exp(-(x^2+y^2)/2)" "6#/-%\"fG6$%\" xG%\"yG*(F'\"\"\"F(F*-%$expG6#,$*&,&*$F'\"\"#F**$F(F2F*F*F2!\"\"F4F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "It is a good idea to make a prelimanary investigation of the function by way of a three dimensional plot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "f := (x,y) \+ -> x*y*exp(-(x^2+y^2)/2);\nplot3d(f(x,y),x=-3..3,y=-3..3,style=patchco ntour,\n color=green,orientation=[67,47],axes=framed,lightmodel=light 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)oper atorG%&arrowGF)*(9$\"\"\"9%F/-%$expG6#,&*$)F.\"\"#F/#!\"\"F7*&#F/F7F/* $)F0F7F/F/F9F/F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT3D 432 272 272 {PLOTDATA 3 "6(-%%GRIDG6&;$!\"$\"\"!$\"\"$F)F&X,%)anythingG6\"6\"[gl'! 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" }}{PARA 0 "" 0 "" {TEXT -1 45 "First the partial derivative with respect to " }{TEXT 273 1 "x" } {TEXT -1 7 " . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Diff(f(x,y),x);\nvalue(%);\nfactor(%);\nf_x := unapply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*(%\" xG\"\"\"%\"yGF(-%$expG6#,&*$)F'\"\"#F(#!\"\"F0*&#F(F0F(*$)F)F0F(F(F2F( F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"yG\"\"\"-%$expG6#,&*$)%\" xG\"\"#F&#!\"\"F.*&#F&F.F&*$)F%F.F&F&F0F&F&*(F,F&F%F&F'F&F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"yG\"\"\"-%$expG6#,&*$)%\"xG\"\"#F&#! \"\"F.*&#F&F.F&*$)F%F.F&F&F0F&,&F-F&F&F0F&,&F-F&F&F&F&F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$f_xGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF), $**9%\"\"\"-%$expG6#,&*$)9$\"\"#F0#!\"\"F8*&#F0F8F0*$)F/F8F0F0F:F0,&F7 F0F0F:F0,&F7F0F0F0F0F:F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 55 " . . . and then the partial derivative wi th respect to " }{TEXT 274 1 "y" }{TEXT -1 7 " . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Diff(f(x, y),y);\nvalue(%);\nfactor(%);\nf_y := unapply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*(%\"xG\"\"\"%\"yGF(-%$expG6#,&*$)F'\"\" #F(#!\"\"F0*&#F(F0F(*$)F)F0F(F(F2F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"-%$expG6#,&*$)F%\"\"#F&#!\"\"F-*&#F&F-F&*$)%\"yGF-F& F&F/F&F&*(F%F&F3F&F'F&F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"xG \"\"\"-%$expG6#,&*$)F%\"\"#F&#!\"\"F-*&#F&F-F&*$)%\"yGF-F&F&F/F&,&F4F& F&F/F&,&F4F&F&F&F&F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f_yGf*6$%\" xG%\"yG6\"6$%)operatorG%&arrowGF),$**9$\"\"\"-%$expG6#,&*$)F/\"\"#F0#! \"\"F7*&#F0F7F0*$)9%F7F0F0F9F0,&F>F0F0F9F0,&F>F0F0F0F0F9F)F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The criti cal points are the points for which f" }{XPPEDIT 18 0 "``[x](x,y)" "6# -&%!G6#%\"xG6$F'%\"yG" }{TEXT -1 6 " and f" }{XPPEDIT 18 0 "``[y](x,y) " "6#-&%!G6#%\"yG6$%\"xGF'" }{TEXT -1 24 " vanish simultaneously." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sols := solve(\{f_x(x,y)=0,f_y(x,y)=0\},\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SOLG6'<$/%\"yG\"\"!/%\"xGF)<$/F+\"\"\"/F(F.<$F- /F(!\"\"<$F//F+F2<$F1F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 48 "Thus we see that there are five critical points:" }{XPPEDIT 18 0 "``(0,0), ``(1,1), ``(1,-1), ``(-1,1)" "6&-%!G6$\"\" !F&-F$6$\"\"\"F)-F$6$F),$F)!\"\"-F$6$,$F)F-F)" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "``(-1,-1)" "6#-%!G6$,$\"\"\"!\"\",$F'F(" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Inspe ction of the plots made above suggest that the points" }{XPPEDIT 18 0 "``(1,1)" "6#-%!G6$\"\"\"F&" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(-1 ,-1)" "6#-%!G6$,$\"\"\"!\"\",$F'F(" }{TEXT -1 31 " are maximum points, and points" }{XPPEDIT 18 0 " ``(-1,1)" "6#-%!G6$,$\"\"\"!\"\"F'" } {TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(1,-1)" "6#-%!G6$\"\"\",$F&!\"\" " }{TEXT -1 21 " are minimum points. " }}{PARA 0 "" 0 "" {TEXT -1 27 " The plots also suggest that" }{XPPEDIT 18 0 " ``(0,0) " "6#-%!G6$\"\" !F&" }{TEXT -1 20 " is a saddle point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "In order to get a clearer picture o f the situation we can plot the surfaces in a neighborhood of each cri tical point. " }}{PARA 0 "" 0 "" {TEXT -1 32 "First look at a neighbou rhood of" }{XPPEDIT 18 0 "``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 2 ". 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xG%\"yGQ!F/-%&STYL EG6#%-PATCHCONTOURG-%+PROJECTIONG6%$\"#q\"\"!$\"#lF?\"\"\"" 1 4 0 1 10 0 2 1 1 1 2 1.000000 65.000000 70.000000 1 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "second derivative test" {TEXT -1 97 "In the same way, \+ one can obtain geometric evidence by plotting around the other critica l points. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Alternatively we can apply the following criteria to characteri ze the critical points." }}{PARA 0 "" 0 "" {TEXT -1 2 "If" }{XPPEDIT 18 0 " ``(a,b) " "6#-%!G6$%\"aG%\"bG" }{TEXT -1 25 " is a critical poi nt of " }{XPPEDIT 18 0 "phi(x,y);" "6#-%$phiG6$%\"xG%\"yG" }{TEXT -1 5 " and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta=`` " "6#/%&DeltaG%!G" }{XPPEDIT 18 0 "Delta(a,b) = ``;" "6#/-%&DeltaG6$% \"aG%\"bG%!G" }{XPPEDIT 18 0 "phi[xx](a,b)*phi[yy](a,b)-phi[xy](a,b)*` `^2;" "6#,&*&-&%$phiG6#%#xxG6$%\"aG%\"bG\"\"\"-&F'6#%#yyG6$F+F,F-F-*&- &F'6#%#xyG6$F+F,F-*$%!G\"\"#F-!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 5 "then:" }}{PARA 15 "" 0 "" {TEXT -1 5 " if " }{XPPEDIT 18 0 "0 < Delta;" "6#2\"\"!%&DeltaG" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "0 < phi[xx](a,b);" "6#2\"\"!-&%$phiG6#%#xxG6$%\"aG%\"bG" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 7 " is a " }{TEXT 259 13 "minimum point" }{TEXT -1 2 ", " }}{PARA 15 "" 0 "" {TEXT -1 5 " if " }{XPPEDIT 18 0 "0 < Delta;" "6#2\"\"!%&D eltaG" }{TEXT -1 11 " > 0 and " }{XPPEDIT 18 0 "phi[xx](a,b) < 0;" " 6#2-&%$phiG6#%#xxG6$%\"aG%\"bG\"\"!" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 7 " is a " }{TEXT 259 13 "maximum point" }{TEXT -1 2 ", " }}{PARA 15 "" 0 "" {TEXT -1 5 " if " }{XPPEDIT 18 0 "Delta < 0;" "6#2%&DeltaG\"\"!" }{TEXT -1 8 ", the n " }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 7 " is a \+ " }{TEXT 259 12 "saddle point" }{TEXT -1 4 ", " }}{PARA 15 "" 0 "" {TEXT -1 5 " if " }{XPPEDIT 18 0 "Delta = 0;" "6#/%&DeltaG\"\"!" } {TEXT -1 66 ", no conclusion can be made about the nature of the criti cal point" }{XPPEDIT 18 0 " ``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 31 " without further investigation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "f := (x,y) -> x*y*exp(-(x^2 +y^2)/2):\nDiff(f(x,y),x$2);\nvalue(%);\nfactor(%):\nf_xx := unapply(% ,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*(%\"xG\"\"\"%\"yG F(-%$expG6#,&*$)F'\"\"#F(#!\"\"F0*&#F(F0F(*$)F)F0F(F(F2F(-%\"$G6$F'F0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(%\"xG\"\"\"%\"yGF&-%$expG6#,&* $)F%\"\"#F&#!\"\"F.*&#F&F.F&*$)F'F.F&F&F0F&!\"$*()F%\"\"$F&F'F&F(F&F& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%f_xxGf*6$%\"xG%\"yG6\"6$%)opera torG%&arrowGF)**9$\"\"\"9%F/-%$expG6#,&*$)F.\"\"#F/#!\"\"F7*&#F/F7F/*$ )F0F7F/F/F9F/,&!\"$F/F5F/F/F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Diff(f(x,y),x,y);\nvalue(%); \nfactor(%):\nf_xy := unapply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6%*(%\"xG\"\"\"%\"yGF(-%$expG6#,&*$)F'\"\"#F(#!\"\"F0*&#F(F 0F(*$)F)F0F(F(F2F(F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%$expG6#, &*$)%\"xG\"\"#\"\"\"#!\"\"F+*&#F,F+F,*$)%\"yGF+F,F,F.F,*&F2F,F$F,F.*&F )F,F$F,F.*(F)F,F2F,F$F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%f_xyGf *6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*,-%$expG6#,&*$)9$\"\"#\"\"\"#! \"\"F5*&#F6F5F6*$)9%F5F6F6F8F6,&F=F6F6F8F6,&F=F6F6F6F6,&F4F6F6F8F6,&F4 F6F6F6F6F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "Diff(f(x,y),y$2);\nvalue(%);\nfactor(%):\nf_yy := unapply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*(%\" xG\"\"\"%\"yGF(-%$expG6#,&*$)F'\"\"#F(#!\"\"F0*&#F(F0F(*$)F)F0F(F(F2F( -%\"$G6$F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(%\"xG\"\"\"%\"yGF& -%$expG6#,&*$)F%\"\"#F&#!\"\"F.*&#F&F.F&*$)F'F.F&F&F0F&!\"$*(F%F&)F'\" \"$F&F(F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%f_yyGf*6$%\"xG%\"yG6 \"6$%)operatorG%&arrowGF)**9$\"\"\"9%F/-%$expG6#,&*$)F.\"\"#F/#!\"\"F7 *&#F/F7F/*$)F0F7F/F/F9F/,&!\"$F/F " 0 "" {MPLTEXT 1 0 68 "f_xx(x,y)*f_ yy(x,y)-f_xy(x,y)^2;\nfactor(%):\nDelta := unapply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*,)%\"xG\"\"#\"\"\")%\"yGF'F()-%$expG6#,&* $F%F(#!\"\"F'*&#F(F'F(*$F)F(F(F2F'F(,&!\"$F(F0F(F(,&F7F(F5F(F(F(*,F+F( ),&F*F(F(F2F'F(),&F*F(F(F(F'F(),&F&F(F(F2F'F(),&F&F(F(F(F'F(F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaGf*6$%\"xG%\"yG6\"6$%)operator G%&arrowGF),$*&)-%$expG6#,&*$)9$\"\"#\"\"\"#!\"\"F7*&#F8F7F8*$)9%F7F8F 8F:F7F8,2*&F5F8F>F8!\"&*&)F?\"\"%F8F5F8F8*&)F6FEF8F>F8F8F8F8*$FGF8F8*$ FDF8F8*&F7F8F5F8F:*&F7F8F>F8F:F8F:F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We first apply the test for the cr itical point" }{XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Delta(0,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus \+ " }{XPPEDIT 18 0 " Delta" "6#%&DeltaG" }{TEXT -1 17 " < 0 at the point " }{XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 21 ", so this point is a " }{TEXT 280 6 "saddle" }{TEXT -1 8 " point. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Now we check the v alues of f" }{XPPEDIT 18 0 "``[xx];" "6#&%!G6#%#xxG" }{TEXT -1 7 " a nd " }{XPPEDIT 18 0 "Delta" "6#%&DeltaG" }{TEXT -1 14 " at the point " }{XPPEDIT 18 0 " ``(1,1)" "6#-%!G6$\"\"\"F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "[f_xx(1,1),simplify(Delta(1,1))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$-%$expG6#!\"\"!\"#,$-F&6#F)\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Since f" }{XPPEDIT 18 0 "``[xx](1, 1);" "6#-&%!G6#%#xxG6$\"\"\"F)" }{TEXT -1 12 " < 0, and " }{XPPEDIT 18 0 "Delta(1,1);" "6#-%&DeltaG6$\"\"\"F&" }{TEXT -1 28 " > 0 at (1,1 ), we see that" }{XPPEDIT 18 0 " ``(1,1)" "6#-%!G6$\"\"\"F&" }{TEXT -1 6 " is a " }{TEXT 279 7 "maximum" }{TEXT -1 8 " point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Next we check the critical point" }{XPPEDIT 18 0 " ``(-1,1)" "6#-%!G6$,$\"\"\"!\"\"F' " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "subs(x=-1,y=1,[fxx,Delta]);\n[f_xx(-1,1),simp lify(Delta(-1,1))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%$fxxG%&Delta G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$-%$expG6#!\"\"\"\"#,$-F&6#!\" #\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }{XPPEDIT 18 0 " ``(-1,1)" "6#-%!G6$,$\"\"\"!\"\"F'" }{TEXT -1 6 " is a " }{TEXT 278 7 "minimum" }{TEXT -1 8 " point. " }}{PARA 0 "" 0 "" {TEXT -1 32 "Next we check the critical point" }{XPPEDIT 18 0 "``(-1,-1);" "6#-%!G6$,$\"\"\"!\"\",$F'F(" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "[f_xx (-1,-1),simplify(Delta(-1,-1))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$ ,$-%$expG6#!\"\"!\"#,$-F&6#F)\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }{XPPEDIT 18 0 "``(-1,-1);" "6#-%!G 6$,$\"\"\"!\"\",$F'F(" }{TEXT -1 6 " is a " }{TEXT 276 7 "maximum" } {TEXT -1 7 " point." }}{PARA 0 "" 0 "" {TEXT -1 35 "Finally we check t he critical point" }{XPPEDIT 18 0 "``(1,-1);" "6#-%!G6$\"\"\",$F&!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "[f_xx(1,-1),simplify(Delta(1,-1))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$-%$expG6#!\"\"\"\"#,$-F&6#!\"#\"\"%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" } {XPPEDIT 18 0 " `` (1,-1)" "6#-%!G6$\"\"\",$F&!\"\"" }{TEXT -1 6 " is \+ a " }{TEXT 277 7 "minimum" }{TEXT -1 7 " point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We can collect all the re quired information regarding the sign of f" }{XPPEDIT 18 0 "``[x,y]" " 6#&%!G6$%\"xG%\"yG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Delta" "6#%&De ltaG" }{TEXT -1 26 ", together with values of " }{XPPEDIT 18 0 "f(x,y) " "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 56 " at the four critical points us ing the Maple procedures " }{TEXT 0 6 "signum" }{TEXT -1 5 " and " } {TEXT 0 3 "zip" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "xvals := [0,1,-1,-1,1]:\nyv als := [0,1,1,-1,-1]:\nzip((x,y)->``(x,y),xvals,yvals);\nzip(signum@f_ xx,xvals,yvals);\nzip(signum@Delta,xvals,yvals);\nzip(f,xvals,yvals); \nevalf(%,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'-%!G6$\"\"!F'-F%6$ \"\"\"F*-F%6$!\"\"F*-F%6$F-F--F%6$F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!!\"\"\"\"\"F%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'!\" \"\"\"\"F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!-%$expG6#!\" \",$F%F(F%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'$\"\"!F%$\"&)yO!\"&$ !&)yOF(F&F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The critical points are classified as follows: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " ``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 9 " f" }{XPPEDIT 18 0 " ``[xx](a,b);" "6#-&%!G6#%#xxG6$%\"aG%\"bG" }{TEXT -1 9 " " } {XPPEDIT 18 0 "Delta" "6#%&DeltaG" }{TEXT -1 15 " " } {TEXT 275 10 "Conclusion" }{TEXT -1 22 " " } {XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 30 " 0 " }{XPPEDIT 18 0 "``-``" "6#,&%!G\" \"\"F$!\"\"" }{TEXT -1 48 " Saddle Point \+ 0" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(1,1)" "6# -%!G6$\"\"\"F&" }{TEXT -1 9 " " }{XPPEDIT 18 0 "``-``" "6#,&%! G\"\"\"F$!\"\"" }{TEXT -1 17 " " }{XPPEDIT 18 0 "``+`` " "6#,&%!G\"\"\"F$F%" }{TEXT -1 48 " Maximum \+ .36788" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``( -1,1)" "6#-%!G6$,$\"\"\"!\"\"F'" }{TEXT -1 7 " " }{XPPEDIT 18 0 "``+``" "6#,&%!G\"\"\"F$F%" }{TEXT -1 17 " " } {XPPEDIT 18 0 "``+``" "6#,&%!G\"\"\"F$F%" }{TEXT -1 49 " Mi nimum -.36788" }}{PARA 15 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "``(-1,-1)" "6#-%!G6$,$\"\"\"!\"\",$F'F(" }{TEXT -1 6 " " }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 16 " " }{XPPEDIT 18 0 "``+`` " "6#,&%!G\"\"\"F$F%" }{TEXT -1 50 " Maximum .36788 " }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(1,-1)" "6#-%!G6$\"\"\",$F&! \"\"" }{TEXT -1 7 " " }{XPPEDIT 18 0 "``+``" "6#,&%!G\"\"\"F$F% " }{TEXT -1 16 " " }{XPPEDIT 18 0 "``+``" "6#,&%!G\"\" \"F$F%" }{TEXT -1 50 " Minimum -.367 88" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "Consider the function f defined by" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x,y) = 3*x^5-3*x*y^2+y^3+3*y^2;" "6# /-%\"fG6$%\"xG%\"yG,**&\"\"$\"\"\"*$F'\"\"&F,F,*(F+F,F'F,F(\"\"#!\"\"* $F(F+F,*&F+F,*$F(F0F,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "f := (x,y) -> 3*x^5-3* x*y^2+y^3+3*y^2;\nplot3d(f(x,y),x=-2.5..1,y=-8..2,style=patchcontour, \n color=red,orientation=[-30,52],axes=boxed,view=-20..40,\n \+ lightmodel=light4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"x G%\"yG6\"6$%)operatorG%&arrowGF),**$)9$\"\"&\"\"\"\"\"$*(F3F2)9%\"\"#F 2F0F2!\"\"*$)F6F3F2F2*&F3F2F5F2F2F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6)-%%GRIDG6&;$!#D!\"\"$\"\"\"\"\"!;$!\")F,$\" \"#F,X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\":\":C0609F0000000000C05F4F5ED 097B420C05E70F684BDA130C05E870000000004C05F75B425ED097EC06090A5ED097B4 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" }}{PARA 0 "" 0 "" {TEXT -1 45 "First the partial derivative \+ with respect to " }{TEXT 291 1 "x" }{TEXT -1 5 " and " }{TEXT 293 1 "y " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f_x := D[1](f);\nf_y := D[2](f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f_xGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF) ,&*$)9$\"\"%\"\"\"\"#:*&\"\"$F2)9%\"\"#F2!\"\"F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f_yGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(*&9 %\"\"\"9$F0!\"'*&\"\"$F0)F/\"\"#F0F0*&\"\"'F0F/F0F0F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The critical poin ts are the points for which f" }{XPPEDIT 18 0 "``[x](x,y)" "6#-&%!G6#% \"xG6$F'%\"yG" }{TEXT -1 6 " and f" }{XPPEDIT 18 0 "``[y](x,y)" "6#-&% !G6#%\"yG6$%\"xGF'" }{TEXT -1 15 " are both zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "solve(\{f_x (x,y)=0,f_y(x,y)=0\},\{x,y\}):\nmap(allvalues,\{%\});\nevalf(%);\nsols := remove(has,%,I);\nxvals := [seq(subs(sols[i],x),i=1..nops(sols))]; \nyvals := [seq(subs(sols[i],y),i=1..nops(sols))];" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#<'<$/%\"yG,(*$-%%sqrtG6#\"\"&\"\"\"#!\"#F,*&#\"\"#F,F -*$-F*6#,&F,F-*&\"#5F-F)F-F-F-F-!\"\"F2F9/%\"xG,&F(#F9F,*&#F-F,F-F3F-F 9<$/F&,(F(F.*&#F2F,F-F4F-F-F2F9/F;,&F(F=*&#F-F,F-F4F-F-<$/F&,(F(FD*&#F 2F,F-*$-F*6#,&F,F-*&F8F-F)F-F9F-F-F9F2F9/F;,&F(FH*&#F-F,F-FNF-F9<$/F&, (F(FD*&FDF-FOF-F-F2F9/F;,&F(FH*&FHF-FOF-F-<$/F&\"\"!/F;Fjn" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<'<$/%\"yG^$$!+4Gd06!\"*$\"+jfkm;F*/%\"xG^$$ \"+cf8sW!#5$\"+9)HKL)F2<$/F.$!+QBO$\\\"F*/F&$!+wYs')\\F*<$/F.$\"+k9N*) fF2/F&$!*1(H@!)F*<$/F.^$F0$!+9)HKL)F2/F&^$F($!+jfkm;F*<$/F&$\"\"!FO/F. FN" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG<%<$/%\"xG$!+QBO$\\\"!\" */%\"yG$!+wYs')\\F+<$/F($\"+k9N*)f!#5/F-$!*1(H@!)F+<$/F-$\"\"!F;/F(F: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7%$!+QBO$\\\"!\"*$\"+k9N* )f!#5$\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7%$!+wYs')\\ !\"*$!*1(H@!)F($\"\"!F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Thus we see that there are three critical points: " }{XPPEDIT 18 0 "``(0,0 )" "6#-%!G6$\"\"!F&" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "``((sqrt(5+10*sq rt(5))-sqrt(5))/5,-2/5*sqrt(5)+2/5*sqrt(5+10*sqrt(5))-2);" "6#-%!G6$*& ,&-%%sqrtG6#,&\"\"&\"\"\"*&\"#5F--F)6#F,F-F-F--F)6#F,!\"\"F-F,F4,(*(\" \"#F-F,F4-F)6#F,F-F4*(F7F-F,F4-F)6#,&F,F-*&F/F--F)6#F,F-F-F-F-F7F4" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "and " }{XPPEDIT 18 0 "` `(-(sqrt(5+10*sqrt(5))+sqrt(5))/5,-2/5*sqrt(5)-2/5*sqrt(5+10*sqrt(5))- 2)" "6#-%!G6$,$*&,&-%%sqrtG6#,&\"\"&\"\"\"*&\"#5F.-F*6#F-F.F.F.-F*6#F- F.F.F-!\"\"F5,(*(\"\"#F.F-F5-F*6#F-F.F5*(F8F.F-F5-F*6#,&F-F.*&F0F.-F*6 #F-F.F.F.F5F8F5" }{TEXT -1 11 ", that is, " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "``(0,0), ``(.5989351464, -.802129706 )" "6$-%!G6$\"\"!F &-F$6$-%&FloatG6$\"+k9N*)f!#5,$-F*6$\"*1(H@!)!\"*!\"\"" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 " ``( -1.493362338, -4.986724676)" "6#-%!G6$,$-%& FloatG6$\"+QBO$\\\"!\"*!\"\",$-F(6$\"+wYs')\\F+F," }{TEXT -1 16 " appr oximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Inspection of the plots made above suggest that " }{XPPEDIT 18 0 "``(-1.493362338,-4.986724676)" "6#-%!G6$,$-%&FloatG6$\"+QBO$\\\"!\" *!\"\",$-F(6$\"+wYs')\\F+F," }{TEXT -1 26 " is a local maximum point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "We ca n apply the following standard criteria to characterize the critical p oints." }}{PARA 0 "" 0 "" {TEXT -1 2 "If" }{XPPEDIT 18 0 " ``(a,b) " " 6#-%!G6$%\"aG%\"bG" }{TEXT -1 25 " is a critical point of " } {XPPEDIT 18 0 "phi(x,y);" "6#-%$phiG6$%\"xG%\"yG" }{TEXT -1 5 " and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta=``" "6#/%&Del taG%!G" }{XPPEDIT 18 0 "Delta(a,b) = ``;" "6#/-%&DeltaG6$%\"aG%\"bG%!G " }{XPPEDIT 18 0 "phi[xx](a,b)*phi[yy](a,b)-phi[xy](a,b)*``^2;" "6#,&* &-&%$phiG6#%#xxG6$%\"aG%\"bG\"\"\"-&F'6#%#yyG6$F+F,F-F-*&-&F'6#%#xyG6$ F+F,F-*$%!G\"\"#F-!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 5 "then:" }}{PARA 15 "" 0 "" {TEXT -1 5 " if " }{XPPEDIT 18 0 "0 < De lta;" "6#2\"\"!%&DeltaG" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "0 < ph i[xx](a,b);" "6#2\"\"!-&%$phiG6#%#xxG6$%\"aG%\"bG" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 7 " is a \+ " }{TEXT 259 13 "minimum point" }{TEXT -1 2 ", " }}{PARA 15 "" 0 "" {TEXT -1 5 " if " }{XPPEDIT 18 0 "0 < Delta;" "6#2\"\"!%&DeltaG" } {TEXT -1 8 " and " }{XPPEDIT 18 0 "phi[xx](a,b) < 0;" "6#2-&%$phiG6 #%#xxG6$%\"aG%\"bG\"\"!" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "``(a,b) " "6#-%!G6$%\"aG%\"bG" }{TEXT -1 7 " is a " }{TEXT 259 13 "maximum po int" }{TEXT -1 2 ", " }}{PARA 15 "" 0 "" {TEXT -1 5 " if " }{XPPEDIT 18 0 "Delta < 0;" "6#2%&DeltaG\"\"!" }{TEXT -1 8 ", then " }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 7 " is a " }{TEXT 259 12 "saddle point" }{TEXT -1 4 ", " }}{PARA 15 "" 0 "" {TEXT -1 5 " i f " }{XPPEDIT 18 0 "Delta = 0;" "6#/%&DeltaG\"\"!" }{TEXT -1 66 ", no conclusion can be made about the nature of the critical point" } {XPPEDIT 18 0 " ``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 31 " without f urther investigation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "f := (x,y) -> 3*x^5-3*x*y^2+y^3+3*y^2:\nf _xx := D[1,1](f);\nf_xy := D[2,1](f);\nf_yy := D[2,2](f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%f_xxGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF ),$*$)9$\"\"$\"\"\"\"#gF)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%f_ xyGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$9%!\"'F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%f_yyGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF ),(9$!\"'*&\"\"'\"\"\"9%F2F2F1F2F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f_xx(x,y)*f_yy(x,y)-f_x y(x,y)^2;\nDelta := unapply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,&*&)%\"xG\"\"$\"\"\",(F&!\"'*&\"\"'F(%\"yGF(F(F,F(F(\"#g*&\"#OF()F- \"\"#F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaGf*6$%\"xG%\"y G6\"6$%)operatorG%&arrowGF),&*&)9$\"\"$\"\"\",(F0!\"'*&\"\"'F29%F2F2F6 F2F2\"#g*&\"#OF2)F7\"\"#F2!\"\"F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "We ca n collect all the required information regarding the sign of f" } {XPPEDIT 18 0 "``[x,y]" "6#&%!G6$%\"xG%\"yG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "Delta" "6#%&DeltaG" }{TEXT -1 26 ", together with value s of " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 57 " a t the three critical points using the Maple procedures " }{TEXT 0 6 "s ignum" }{TEXT -1 5 " and " }{TEXT 0 3 "zip" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "z ip((x,y)->``(x,y),xvals,yvals);\nzip(signum@f_xx,xvals,yvals);\nzip(si gnum@Delta,xvals,yvals);\nzip(f,xvals,yvals);\nevalf(%,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%!G6$$!+QBO$\\\"!\"*$!+wYs')\\F)-F%6$$\"+ k9N*)f!#5$!*1(H@!)F)-F%6$$\"\"!F6F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7%!\"\"\"\"\"$\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"!\" \"$\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$\"+5_=sR!\")$\"*FsE*[ !\"*$\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$\"&A(R!\"$$\"&F*[! \"&$\"\"!F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "This information can be tabulated as follows: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` (a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 50 " \+ f" }{XPPEDIT 18 0 "``[xx](a,b);" "6#-&%!G6#%#xxG6 $%\"aG%\"bG" }{TEXT -1 9 " " }{XPPEDIT 18 0 "Delta" "6#%&Delta G" }{TEXT -1 15 " " }{TEXT 292 10 "Conclusion" }{TEXT -1 22 " " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"x G%\"yG" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(0,0)" "6 #-%!G6$\"\"!F&" }{TEXT -1 73 " \+ 0 " }{XPPEDIT 18 0 "0;" "6#\"\"!" } {TEXT -1 55 " Test fails 0" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(.5989351464,-.80 2129706)" "6#-%!G6$-%&FloatG6$\"+k9N*)f!#5,$-F'6$\"*1(H@!)!\"*!\"\"" } {TEXT -1 12 " " }{XPPEDIT 18 0 "``+``" "6#,&%!G\"\"\"F$F%" }{TEXT -1 16 " " }{XPPEDIT 18 0 "``-``;" "6#,&%!G\"\"\" F$!\"\"" }{TEXT -1 52 " Saddle point . 48927" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(-1.493362 338,-4.986724676)" "6#-%!G6$,$-%&FloatG6$\"+QBO$\\\"!\"*!\"\",$-F(6$\" +wYs')\\F+F," }{TEXT -1 8 " " }{XPPEDIT 18 0 "``-``;" "6#,&%!G \"\"\"F$!\"\"" }{TEXT -1 16 " " }{XPPEDIT 18 0 "``+``" "6#,&%!G\"\"\"F$F%" }{TEXT -1 51 " Maximum \+ 39.722" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "To determine the nature of the critical point" }{XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 52 ", we can zoom in with \+ a 3d plot and a contour plot. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "f := (x,y) -> 3*x^5-3*x*y^2 +y^3+3*y^2;\nplot3d(f(x,y),x=-0.2..0.2,y=-0.02..0.02,style=patchcontou r,\n color=red,lightmodel=light4,axes=framed);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),**$)9$ \"\"&\"\"\"\"\"$*(F3F2F0F2)9%\"\"#F2!\"\"*$)F6F3F2F2*&F3F2F5F2F2F)F)F) " }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%%GRIDG6&;$ !\"#!\"\"$\"\"#F);$F(F($F+F(X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\":\":3F 3EEED8904F6DFC3F2FF5D8272458F43F028B54AC01FA40BF241A6937D1FE6CBF35207A 32AC34A4BF3EE7EEE0CC069CBF43B18DAC258D5CBF4748C425960B4BBF4A395F35B8CF EABF4C8323358F2E0ABF4E25D47E1A7874BF4F2137685C01FBBF4F75104D551D6BBF4F 212386071D94BF4E25356B735545BF4C810A569B1749BF4A3466A07FB672BF473F0EA2 22858EBF43A0C6B484D76ABF3EB2A6614FFDABBF34D0F0DF1A9D40BF2337EB28D86658 3F07661665C4D7903F30C9AF9251277E3F3FFB480A5ACCDC3F49E88C237CC2463F4289 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Examp le 4 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 34 "Consider the function f defined by" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x,y)=exp(-(x^2+y^2)/2)*(x^2-x*y+y^2 )" "6#/-%\"fG6$%\"xG%\"yG*&-%$expG6#,$*&,&*$F'\"\"#\"\"\"*$F(F1F2F2F1! \"\"F4F2,(*$F'F1F2*&F'F2F(F2F4*$F(F1F2F2" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "f := (x,y) -> exp(-(x^2+y^2)/2)*(x^2-x*y+y^2);\nplot3d(f(x,y),x=-3.2..3.2, y=-3.2..3.2,grid=[30,30],\n style=patchcontour,color=coral,orientatio n=[22,26],\n axes=boxed,lightmodel=light4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&-%$expG 6#,&*$)9$\"\"#\"\"\"#!\"\"F5*&#F6F5F6*$)9%F5F6F6F8F6,(F2F6*&F4F6F=F6F8 F;F6F6F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT3D 437 255 255 {PLOTDATA 3 "6( -%%GRIDG6&;$!#K!\"\"$\"#KF)F&X,%)anythingG6\"6\"[gl'!%\"!!#_cn\"?\"?3F 37F76BBCD8948A3F462D00225D48843F53A9EBCD57D8A03F60BBDD2C6013C43F6B5F8A C04240403F758BF274978EA83F80574346CBEE123F87E625622355023F90DB75CCF580 BD3F96EF7379F762D83F9E1414216652E33FA2FE24E228AF943FA711922650CC113FAA E8FA962CA38E3FAE1AC6833DF80E3FB0204A88BE2A093FB085FEDB5F2F143FB02C73C8 22475A3FAE378D086B35EC3FAAEA76B5B4D5FC3FA6D89F6B5BBFD23FA277EAD0FA5D21 3F9C6CCFF658ADF63F94D13A2D616F003F8D029DEF8F1A513F833A75B8B164193F783E 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/%\"xGF)<$/F+\"\"\"/F(!\"\"<$/F(F./F+F0<$F2F-<$F3F/" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&xvalsG7'\"\"!\"\"\"!\"\"F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7'\"\"!!\"\"\"\"\"F(F'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 49 "Thus we see that there are five critical points: \+ " }{XPPEDIT 18 0 "``(0,0),``(1,-1),``(-1,1),``(1,1)" "6&-%!G6$\"\"!F&- F$6$\"\"\",$F)!\"\"-F$6$,$F)F+F)-F$6$F)F)" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "``(-1,-1)" "6#-%!G6$,$\"\"\"!\"\",$F'F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Insp ection of the plots made above suggest that" }{XPPEDIT 18 0 " ``(0,0) " "6#-%!G6$\"\"!F&" }{TEXT -1 19 " is a minimum point" }{XPPEDIT 18 0 "``(1,-1)" "6#-%!G6$\"\"\",$F&!\"\"" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(-1,1)" "6#-%!G6$,$\"\"\"!\"\"F'" }{TEXT -1 24 " are maximum poi nts and " }{XPPEDIT 18 0 "``(1,1)" "6#-%!G6$\"\"\"F&" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "``(-1,-1)" "6#-%!G6$,$\"\"\"!\"\",$F'F(" }{TEXT -1 20 " are saddle points. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "We can apply the following standard criteria to characterize the critical points." }}{PARA 0 "" 0 "" {TEXT -1 2 "If" }{XPPEDIT 18 0 " ``(a,b) " "6#-%!G6$%\"aG%\"bG" }{TEXT -1 25 " is a cr itical point of " }{XPPEDIT 18 0 "phi(x,y);" "6#-%$phiG6$%\"xG%\"yG" }{TEXT -1 5 " and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta=``" "6#/%&DeltaG%!G" }{XPPEDIT 18 0 "Delta(a,b) = ``;" "6#/-%&D eltaG6$%\"aG%\"bG%!G" }{XPPEDIT 18 0 "phi[xx](a,b)*phi[yy](a,b)-phi[xy ](a,b)*``^2;" "6#,&*&-&%$phiG6#%#xxG6$%\"aG%\"bG\"\"\"-&F'6#%#yyG6$F+F ,F-F-*&-&F'6#%#xyG6$F+F,F-*$%!G\"\"#F-!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 5 "then:" }}{PARA 15 "" 0 "" {TEXT -1 5 " if " } {XPPEDIT 18 0 "0 < Delta;" "6#2\"\"!%&DeltaG" }{TEXT -1 8 " and " } {XPPEDIT 18 0 "0 < phi[xx](a,b);" "6#2\"\"!-&%$phiG6#%#xxG6$%\"aG%\"bG " }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 7 " is a " }{TEXT 259 13 "minimum point" }{TEXT -1 2 ", " } }{PARA 15 "" 0 "" {TEXT -1 5 " if " }{XPPEDIT 18 0 "0 < Delta;" "6#2 \"\"!%&DeltaG" }{TEXT -1 11 " > 0 and " }{XPPEDIT 18 0 "phi[xx](a,b) < 0;" "6#2-&%$phiG6#%#xxG6$%\"aG%\"bG\"\"!" }{TEXT -1 7 ", then " } {XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 7 " is a " } {TEXT 259 13 "maximum point" }{TEXT -1 2 ", " }}{PARA 15 "" 0 "" {TEXT -1 5 " if " }{XPPEDIT 18 0 "Delta < 0;" "6#2%&DeltaG\"\"!" } {TEXT -1 8 ", then " }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" } {TEXT -1 7 " is a " }{TEXT 259 12 "saddle point" }{TEXT -1 4 ", " } }{PARA 15 "" 0 "" {TEXT -1 5 " if " }{XPPEDIT 18 0 "Delta = 0;" "6#/% &DeltaG\"\"!" }{TEXT -1 66 ", no conclusion can be made about the natu re of the critical point" }{XPPEDIT 18 0 " ``(a,b)" "6#-%!G6$%\"aG%\"b G" }{TEXT -1 31 " without further investigation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "f := (x,y) \+ -> exp(-(x^2+y^2)/2)*(x^2-x*y+y^2):\nf_xx := unapply(factor(D[1,1](f)( x,y)),x,y);\nf_xy := unapply(factor(D[2,1](f)(x,y)),x,y);\nf_yy := una pply(factor(D[2,2](f)(x,y)),x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%f_xxGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&-%$expG6#,&*$)9$\"\"# \"\"\"#!\"\"F5*&#F6F5F6*$)9%F5F6F6F8F6,0F2!\"&*(\"\"$F6F4F6F=F6F6F;F8* $)F4\"\"%F6F6*&)F4FAF6F=F6F8*&F3F6F%%f_xyGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&-% $expG6#,&*$)9$\"\"#\"\"\"#!\"\"F5*&#F6F5F6*$)9%F5F6F6F8F6,0*&)F4\"\"$F 6F=F6F6*&F3F6F%%f_yyGf*6$%\"xG%\"yG6\"6$%)op eratorG%&arrowGF)*&-%$expG6#,&*$)9$\"\"#\"\"\"#!\"\"F5*&#F6F5F6*$)9%F5 F6F6F8F6,0F2F8*(\"\"$F6F4F6F=F6F6*&\"\"&F6F " 0 "" {MPLTEXT 1 0 68 "f_xx(x,y)*f_yy(x,y)-f_xy(x,y )^2:\nfactor(%):\nDelta := unapply(%,x,y);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&DeltaGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&)- %$expG6#,&*$)9$\"\"#\"\"\"#!\"\"F7*&#F8F7F8*$)9%F7F8F8F:F7F8,B*&)F6\" \"&F8F?F8!\"#*(\"\"%F8F6F8F?F8F:*(F7F8F6F8)F?FCF8F:*(\"#5F8F6F8)F?\"\" $F8F8*(FFF8F5F8)F?FFF8F8*(FFF8)F6FLF8FKF8F:*(FFF8)F6FFF8F>F8F8*(\"#>F8 F5F8F>F8F:*(FJF8FPF8F?F8F8*&\"\"'F8FRF8F:*&FJF8F5F8F8*&FWF8FNF8F:*$)F? FWF8F8*&FJF8F>F8F8*$)F6FWF8F8FLF:F8F:F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "We can collect all the required information regarding the sign of f " }{XPPEDIT 18 0 "``[x,y]" "6#&%!G6$%\"xG%\"yG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "Delta" "6#%&DeltaG" }{TEXT -1 26 ", together with value s of " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 57 " a t the three critical points using the Maple procedures " }{TEXT 0 6 "s ignum" }{TEXT -1 5 " and " }{TEXT 0 3 "zip" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "z ip((x,y)->``(x,y),xvals,yvals);\nzip(signum@f_xx,xvals,yvals);\nzip(si gnum@Delta,xvals,yvals);\nzip(f,xvals,yvals);\nevalf(%,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'-%!G6$\"\"!F'-F%6$\"\"\"!\"\"-F%6$F+F*-F%6 $F*F*-F%6$F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"\"!\"\"F%\"\"! F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"\"F$F$!\"\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!,$-%$expG6#!\"\"\"\"$F%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'$\"\"!F%$\"&O5\"!\"%F&$\"&)yO!\"&F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "This info rmation can be tabulated as follows: " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6 $%\"aG%\"bG" }{TEXT -1 15 " f" }{XPPEDIT 18 0 "``[xx](a,b );" "6#-&%!G6#%#xxG6$%\"aG%\"bG" }{TEXT -1 9 " " }{XPPEDIT 18 0 "Delta" "6#%&DeltaG" }{TEXT -1 17 " " }{TEXT 289 10 "Conclusion" }{TEXT -1 22 " " }{XPPEDIT 18 0 "f(x ,y)" "6#-%\"fG6$%\"xG%\"yG" }}{PARA 15 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 17 " \+ " }{XPPEDIT 18 0 "``+``" "6#,&%!G\"\"\"F$F%" }{TEXT -1 13 " \+ " }{XPPEDIT 18 0 "``+``;" "6#,&%!G\"\"\"F$F%" }{TEXT -1 52 " \+ Minimum 0" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(1,-1);" "6#-%!G6$\"\"\",$F&!\"\"" } {TEXT -1 16 " " }{XPPEDIT 18 0 "``-``;" "6#,&%!G\"\"\"F $!\"\"" }{TEXT -1 13 " " }{XPPEDIT 18 0 "``+``;" "6#,&%!G \"\"\"F$F%" }{TEXT -1 50 " Maximum \+ 1.1036" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(-1, 1); " "6#-%!G6$,$\"\"\"!\"\"F'" }{TEXT -1 16 " " }{XPPEDIT 18 0 "``-``;" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 13 " " } {XPPEDIT 18 0 "``+``;" "6#,&%!G\"\"\"F$F%" }{TEXT -1 50 " \+ Maximum 1.1036" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(1,1);" "6#-%!G6$\"\"\"F&" }{TEXT -1 36 " \+ 0 " }{XPPEDIT 18 0 "``-``;" "6#,&%!G\"\"\"F $!\"\"" }{TEXT -1 51 " Saddle point .36 788" }}{PARA 15 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(-1, -1);" " 6#-%!G6$,$\"\"\"!\"\",$F'F(" }{TEXT -1 33 " 0 \+ " }{XPPEDIT 18 0 "``-``;" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 51 " \+ Saddle point .36788" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 " Q1 " }}{PARA 0 "" 0 "" {TEXT -1 49 "(a) Find all the critical points o f the function " }{XPPEDIT 18 0 "g(x,y) = x^3+x*y+y^2;" "6#/-%\"gG6$% \"xG%\"yG,(*$F'\"\"$\"\"\"*&F'F,F(F,F,*$F(\"\"#F," }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 109 "(b) Determine whether each critical poin t is a local maximum point, a local minimum point or a saddle point. \+ " }}{PARA 0 "" 0 "" {TEXT -1 47 "(c) Plot a 3 dimensional graph of the function " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 38 ", which shows all the critical points." }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }} {PARA 0 "" 0 "" {TEXT -1 49 "(a) Find all the critical points of the f unction " }{XPPEDIT 18 0 "g(x,y) = exp(x+y)*(x^2-x*y+y^2);" "6#/-%\"gG 6$%\"xG%\"yG*&-%$expG6#,&F'\"\"\"F(F.F.,(*$F'\"\"#F.*&F'F.F(F.!\"\"*$F (F1F.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 109 "(b) Determine whether each critical point is a local maximum point, a local minimum point or a saddle point. " }}{PARA 0 "" 0 "" {TEXT -1 47 "(c) Plot a \+ 3 dimensional graph of the function " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"g G6$%\"xG%\"yG" }{TEXT -1 38 ", which shows all the critical points." } }{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 49 "(a) Find all the cri tical points of the function " }{XPPEDIT 18 0 "g(x,y)=x^4+4*x^2*y^2-2* x^2+2*y^2-1" "6#/-%\"gG6$%\"xG%\"yG,,*$F'\"\"%\"\"\"*(F+F,*$F'\"\"#F,F (F/F,*&F/F,*$F'F/F,!\"\"*&F/F,*$F(F/F,F,F,F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 109 "(b) Determine whether each critical point is a local maximum point, a local minimum point or a saddle point. " }} {PARA 0 "" 0 "" {TEXT -1 47 "(c) Plot a 3 dimensional graph of the fun ction " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 38 ", which shows all the critical points." }}{PARA 0 "" 0 "" {TEXT -1 32 " ________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "__ ______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 " " {TEXT -1 49 "(a) Find all the critical points of the function " } {XPPEDIT 18 0 "g(x,y) = 3/4*y^2+y^3/24-y^4/32-x^2;" "6#/-%\"gG6$%\"xG% \"yG,**(\"\"$\"\"\"\"\"%!\"\"F(\"\"#F,*&F(F+\"#CF.F,*&F(F-\"#KF.F.*$F' F/F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 109 "(b) Determine wh ether each critical point is a local maximum point, a local minimum po int or a saddle point. " }}{PARA 0 "" 0 "" {TEXT -1 47 "(c) Plot a 3 d imensional graph of the function " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$ %\"xG%\"yG" }{TEXT -1 38 ", which shows all the critical points." }} {PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 49 "(a) Find all the cri tical points of the function " }{XPPEDIT 18 0 "g(x,y) = (2*x^2+3*y^2)* exp(-x^2-y^2);" "6#/-%\"gG6$%\"xG%\"yG*&,&*&\"\"#\"\"\"*$F'F,F-F-*&\" \"$F-*$F(F,F-F-F--%$expG6#,&*$F'F,!\"\"*$F(F,F7F-" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 109 "(b) Determine whether each critical poin t is a local maximum point, a local minimum point or a saddle point. \+ " }}{PARA 0 "" 0 "" {TEXT -1 47 "(c) Plot a 3 dimensional graph of the function " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 38 ", which shows all the critical points." }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }} {PARA 0 "" 0 "" {TEXT -1 28 "Construct a 7 digit integer " }{TEXT 262 2 "SN" }{TEXT -1 58 " (using your student number, if you are an RMIT s tudent). " }}{PARA 0 "" 0 "" {TEXT -1 25 "Then set up the constant " } {TEXT 262 5 "sigma" }{TEXT -1 58 " using the following code (with the \+ appropriate value for " }{TEXT 262 2 "SN" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "SN : = 9967057;\nsigma := evalf(SN)*Float(1,-7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 49 "(a) Find all the critical points of the function " }{XPPEDIT 18 0 "g(x,y) = x*(y+sigma *x)*exp(-x^2-y^2);" "6#/-%\"gG6$%\"xG%\"yG*(F'\"\"\",&F(F**&%&sigmaGF* F'F*F*F*-%$expG6#,&*$F'\"\"#!\"\"*$F(F3F4F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 109 "(b) Determine whether each critical point is a local maximum point, a local minimum point or a saddle point. " }} {PARA 0 "" 0 "" {TEXT -1 47 "(c) Plot a 3 dimensional graph of the fun ction " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 38 ", which shows all the critical points." }}{PARA 0 "" 0 "" {TEXT -1 32 " ________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "__ ______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "C ode for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 0 "" 0 "" {TEXT -1 38 "pictures for types of critical po ints " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 299 "p1 := plot3d([r*cos(t),r*sin(t),exp(-r^2)],t=0..2*Pi ,r=0..1.5):\np2 := plottools[arrow]([.6,-.6,1.4],[0,0,1],.05,.12,.12,c ylindrical_arrow,color=red):\nt1 := plots[textplot3d]([.9,-.9,1.5,`max imum point`],color=black,align=BELOW):\nplots[display]([p1,p2,t1],colo r=COLOR(RGB,.7,.7,1),lightmodel=light2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "p1 := plot3d([r*cos( t),r*sin(t),-exp(-r^2)],t=0..2*Pi,r=0..1.5):\np2 := plottools[arrow]([ .6,-.6,-1.4],[0,0,-1],.05,.12,.12,cylindrical_arrow,color=red):\nt1 := plots[textplot3d]([.9,-.9,-1.6,`minimum point`],color=black,align=BEL OW):\nplots[display]([p1,p2,t1],color=COLOR(RGB,.9,.9,1),lightmodel=li ght2,\n orientation=[45,100]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 342 "p1 := plot3d([r*cos(t), r*sin(t),r^2*cos(t)*sin(t)],t=0..2*Pi,r=0..1):\np2 := plottools[arrow] ([-.6,.6,.4],[0,0,0],.05,.12,.12,cylindrical_arrow,color=red):\nt1 := \+ plots[textplot3d]([-.65,.45,.45,`saddle point`],color=black,align=BELO W):\nplots[display]([p1,p2,t1],orientation=[-15,50],\n c olor=COLOR(RGB,1,.8,.8),lightmodel=light2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }