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We divide the problem into several cases. \+ " }{TEXT 259 5 "Expr " }{TEXT -1 130 "is the Euclidian distance betwee n the coordinate center and centroid in each case. It is the expressio n we are trying to minimize." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#%\"cG,&*&&%\"xG6#\" \"!\"\"\"&%\"yGF,F.F.*$),(F*F.F/F.!\"\"F.\"\"#\"\"\"#F4F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xcG,$*&,(*$)&%\"xG6#\"\"!\"\"#\"\"\"\"\"$*$) F*F0F/!\"#*$),&\"\"\"F7&%\"yGF,!\"\"F0F/F:F/,&*&F*F7F8F7F7*$),(F*F7F8F 7F:F7F.F/#F:F.!\"\"#F7\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ycG, $*&,(*$)&%\"yG6#\"\"!\"\"#\"\"\"\"\"$*$)F*F0F/!\"#*$),&\"\"\"F7&%\"xGF ,!\"\"F0F/F:F/,&*&F8F7F*F7F7*$),(F8F7F*F7F:F7F.F/#F:F.!\"\"#F7\"\"'" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 6 "Case \+ I" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "x+y <= 1;" "6#1,&%\"xG\"\"\"%\"yGF &F&" }}{PARA 0 "" 0 "" {TEXT -1 82 "The region is a rectangle and the \+ centers are the same. The maximum distance is 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 8 "Case II:" }{TEXT -1 32 " x+y > \+ 1 and y<1-x/2 and x<1-y/2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG* $-%%sqrtG6#,&*$),&&%\"xG6#\"\"!#\"\"\"\"\"#*&,(*$)F-F3\"\"\"\"\"$*$)F- F9F8!\"#*$),&F2F2&%\"yGF/!\"\"F9F8FBF8,&*&F-F2F@F2F2*$),(F-F2F@F2FBF2F 3F8#FBF3!\"\"#FB\"\"'F3F8F2*$),&F@F1*&,(*$)F@F3F8F9*$)F@F9F8F<*$),&F2F 2F-FBF9F8FBF8FCFIFJF3F8F2F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "p:= proc(x,y) \nif x+y > 1 and y<1-x/2 and x<1-y/2 then \nsqrt((1 /2*x-1/6*(3*x^2-2*x^3-(1-y)^3)/(x*y-1/2*(x+y-1)^2))^2+\n(1/2*y-1/6*(3* y^2-2*y^3-(1-x)^3)/(x*y-1/2*(x+y-1)^2))^2)\nelse 0 \nfi end;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot3d(p,0..1,0..1):" }}{PARA 260 "" 1 " " {TEXT -1 0 "" }{GLPLOT3D 291 214 214 {PLOTDATA 3 "6#-%%GRIDG6%;\"\"! $\"\"\"F'F&W(\\bm\":\":00000000000000000000000000000000000000000000000 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0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000" 1 2 0 1 0 2 1 1 2 2 1.000000 60.000000 -48.000000 0 }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "evalf(solve(\{diff(expr,x[0 ])=0,diff(expr,y[0])=0\},\{x[0],y[0]\}));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$<$/&%\"yG6#\"\"!$\"+GapEx!#6/&%\"xGF'F)<$/F%,&$!+Xz&3Q \"!#5\"\"\"%\"IG$!+4Z7(p%F4/F-,&$!+Wz&3Q\"F4F5F6$\"+4Z7(p%F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ex1:=subs(y[0]=1-x[0]/2,expr ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf(solve(diff(ex1, x[0])=0,x[0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+TOE#)G!\"*,&$\" +%Q=sC&!#5\"\"\"%\"IG$\"+&e4`g%F),&F'F*F+$!+&e4`g%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ex2:=subs(x[0]=1-y[0]/2,expr):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf(solve(diff(ex2,y[0])=0 ,y[0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+TOE#)G!\"*,&$\"+%Q=sC& !#5\"\"\"%\"IG$\"+&e4`g%F),&F'F*F+$!+&e4`g%F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "None of the extrema found so far satisfy x,y<1. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "subs(\{x[0]=0.001,y[0]=.999 \},expr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "subs(\{y[0]=0.001,x[0]=.999\},expr);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs(\{x[0]=2/3,y[0]=2/3\},expr);" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"#\"\"\"#F(\"#j" }}}{EXCHG {PARA 261 "" 1 "" {TEXT -1 24 "but a local extremum of " }{XPPEDIT 18 0 "2/63*sqrt(2);" "6#*(\"\"#\"\"\"\"#j!\"\"-%%sqrtG6#\"\"#F%" }{TEXT -1 25 " is found at [2/3,2/3]. " }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 1 "" {TEXT 256 10 "Case III. " }{TEXT -1 9 "x+y>1 and" } {TEXT 265 1 " " }{XPPEDIT 18 0 "2/3 <= y;" "6#1#\"\"#\"\"$%\"yG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "2/3 <= x;" "6#1#\"\"#\"\"$%\"xG" }} }{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expr2G*$-%%sqrtG6#,&*$),&# \"\"\"\"\"$F.*&,(*$)&%\"xG6#\"\"!\"\"#\"\"\"F/*$)F4F/F9!\"#*$),&F.F.&% \"yGF6!\"\"F/F9FBF9,&*&F4F.F@F.F.*$),(F4F.F@F.FBF.F8F9#FBF8!\"\"#FB\" \"'F8F9F.*$),&F-F.*&,(*$)F@F8F9F/*$)F@F/F9F<*$),&F.F.F4FBF/F9FBF9FCFIF JF8F9F.F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "p2:= proc(x,y ) \nif y >= 2/3 and x>=2/3 then \nsqrt((1/3-1/6*(3*x^2-2*x^3-(1-y)^3)/ (x*y-1/2*(x+y-1)^2))^2+\n(1/3-1/6*(3*y^2-2*y^3-(1-x)^3)/(x*y-1/2*(x+y- 1)^2))^2)\nelse 0 " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot3d(p2,0.6..1,0.6..1);" }}{PARA 13 "" 1 "" {GLPLOT3D 319 229 229 {PLOTDATA 3 "6#-%%GRIDG6%;$\"\"'!\"\"$\"\"\"\"\" !F&W(\\bm\":\":0000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 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"ex1:=subs(y[0]=1-x[0]/2,expr4):\nevalf(solve(diff(ex1,x[0])=0,x[ 0]));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+TOE#)G!\"* ,&$\"+%Q=sC&!#5\"\"\"%\"IG$\"+&e4`g%F),&F'F*F+$!+&e4`g%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "ex2:=subs(x[0]=2/3,expr4):\nevalf(s olve(diff(ex2,y[0])=0,y[0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"\" \"\"\"!$!+fnHWn!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "ex3:= subs(y[0]=1,expr4):\nevalf(solve(diff(ex1,x[0])=0,x[0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+TOE#)G!\"*,&$\"+%Q=sC&!#5\"\"\"%\"IG$\"+&e4 `g%F),&F'F*F+$!+&e4`g%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "subs(\{y[0]=1,x[0]=2/3\},expr4);\nsubs(\{x[0]=2/3,y[0]=2/3\},expr4); \nsubs(\{y[0]=1,x[0]=0.001\},expr4);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$*&-%%sqrtG6#\"\"&\"\"\"-F&6#\"%'H\"F)#\"\"\"F," }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&-%%sqrtG6#\"\")\"\"\"-F&6#\"%pRF)#\"\"\"F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+lu;A$*!#<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "As 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