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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 47 "Constrained Optimization of Biv
ariate Functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart;
\nwith(plots):" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 18 "General Para
meters" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "Set some options for al
l plots. You can change these to your liking (or simply change the set
tings for each individual plot)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 
"setoptions3d(labels=[`x`,`y`, 'z'], axes=frame):" }}}}{SECT 1 {PARA 
257 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
55 "Define the function along with its rectangular domains." }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 81 "f := (x,y) -> x^2 + y;\nxf := x=-2..2:\ny
f := y=-2..2:\ng := (x,y) -> x^2 + y^2 - 1;" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 18 "Plot the graph of " }{XPPEDIT 18 0 "f" "6#%\"fG" }
{TEXT -1 170 " along with the constraint circle (black, its projection
 onto the surface as well as below the surface as a planar level curve
), and the extreme points computed in class:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 602 "fplot := plot3d(f(x,y), xf, yf, style=patchnogrid):
\ng1plot := spacecurve([cos(t),sin(t),-2], t=0..2*Pi, thickness=3, col
or=black):\ng2plot := spacecurve([cos(t),sin(t),f(cos(t),sin(t))], t=0
..2*Pi, thickness=3, color=black):\np1plot := pointplot3d([0,-1,f(0,-1
)], symbolsize=20, symbol=diamond, color=blue, thickness=10):\np2plot \+
:= pointplot3d([sqrt(3)/2,1/2,f(sqrt(3)/2,1/2)], symbolsize=20, symbol
=diamond, color=blue, thickness=10):\np3plot := pointplot3d([-sqrt(3)/
2,1/2,f(-sqrt(3)/2,1/2)], symbolsize=20, symbol=diamond, color=blue, t
hickness=10):\ndisplay(\{fplot,g1plot,g2plot,p1plot,p2plot,p3plot\});
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 257 
"" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "T
his is an illustration of the solution of the following constrained op
timization problem:" }}{PARA 0 "" 0 "" {TEXT -1 9 "Maximize " }
{XPPEDIT 18 0 "f(x,y,z) = x + 2*y + 3*z" "6#/-%\"fG6%%\"xG%\"yG%\"zG,(
F'\"\"\"*&\"\"#F+F(F+F+*&\"\"$F+F)F+F+" }{TEXT -1 34 " on the intersec
tion of the plane " }{XPPEDIT 18 0 "x-y+z=1" "6#/,(%\"xG\"\"\"%\"yG!\"
\"%\"zGF&F&" }{TEXT -1 18 " and the cylinder " }{XPPEDIT 18 0 "x^2+y^2
=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 48 "The plane is colored according to the values of " 
}{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 174 ", so the two extrema (iden
tify with blue markers) occur at the \"hottest\" and \"coldest\" point
s in the plane along the black intersection curve of the cylinder with
 the plane. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 424 "p1 := plot3d(1-x+y
, x=-1..1, y=-1..1, style=patchnogrid, color=x+2*y+3*(1-x+y)):\np2 := \+
spacecurve([cos(t),sin(t),1-cos(t)+sin(t)], t=0..2*Pi, thickness=2, co
lor=black):\np3 := pointplot3d([-2/sqrt(29),5/sqrt(29),1+7/sqrt(29)], \+
symbolsize=20, symbol=diamond, color=blue, thickness=10):\np4 := point
plot3d([2/sqrt(29),-5/sqrt(29),1-7/sqrt(29)], symbolsize=20, symbol=di
amond, color=blue, thickness=10):\ndisplay3d(\{p1,p2,p3,p4\});" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "3 0 0" 9 }
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