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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 38 "Solving Multivariable Max-Min Pr
oblems" }}{PARA 18 "" 0 "" {TEXT -1 20 "Using Maple and the " }{TEXT 
256 8 "vec_calc" }{TEXT -1 8 " Package" }}{PARA 0 "" 0 "" {TEXT -1 83 
"This worksheet gives examples of how to solve multivariable max-min p
roblems using " }{TEXT 265 5 "Maple" }{TEXT -1 9 " and the " }{TEXT 
266 8 "vec_calc" }{TEXT -1 9 " package." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 49 "     There are two main classes of s
uch problems:" }}{PARA 0 "" 0 "" {TEXT 267 28 "* The Unconstrained Pro
blem:" }}{PARA 0 "" 0 "" {TEXT -1 111 "Find all critical points of a f
unction and classify each as a local maximum, a local minimum or a sad
dle point." }}{PARA 0 "" 0 "" {TEXT 268 26 "* The Constrained Problem:
" }}{PARA 0 "" 0 "" {TEXT -1 97 "Find the absolute maximum and minimum
 values of a function inside or on the boundary of a region." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Examples of the
se are given in the collapsed sections below." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "To start the " }{TEXT 
271 8 "vec_calc" }{TEXT -1 41 " package, execute the following command
s:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "libname:=libname,\"C:/
mylib/vector\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(libnameG6$Q=C:\\
Program~Files\\Maple~6/lib6\"Q0C:/mylib/vectorF'" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): vc_aliases:" }}{PARA 7 "" 1 "
" {TEXT -1 80 "Warning, the protected names norm and trace have been r
edefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, th
e name changecoords has been redefined\n" }}{PARA 6 "" 1 "" {TEXT -1 
33 "Package:   vec_calc   Version 4.3" }}{PARA 6 "" 1 "" {TEXT -1 32 "
For all HELP, execute: ?vec_calc" }}{PARA 6 "" 1 "" {TEXT -1 38 "To us
e aliases, execute:   vc_aliases;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 26 "The Unconstrained Problem:" }}{PARA 0 "" 0 "" {TEXT -1 111 "Fin
d all critical points of a function and classify each as a local maxim
um, a local minimum or a saddle point." }}{SECT 0 {PARA 256 "" 0 "" 
{TEXT -1 3 "Exa" }{TEXT 259 0 "" }{TEXT 260 0 "" }{TEXT 261 0 "" }
{TEXT -1 0 "" }{TEXT 262 0 "" }{TEXT 263 0 "" }{TEXT 264 0 "" }{TEXT 
-1 2 "mp" }{TEXT 257 0 "" }{TEXT -1 3 "le:" }}{PARA 0 "" 0 "" {TEXT 
-1 46 "Classify the critical points of the function: " }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "f:=(x,y)->x*y*exp(-x^2/2-y^2/8);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"yG6\"6$%)operatorG%&a
rrowGF)*(9$\"\"\"9%F/-%$expG6#,&*$)F.\"\"#F/#!\"\"F7*&#F/\"\")F/*$)F0F
7F/F/F9F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT 258 9 "Solution:" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 83 "Compute the gradient of f, set it equal t
o zero and solve for the critical points: " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "delf:=GRAD(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%%delfG7$R6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF*,&*&9%\"\"\"-%$expG6#,
&*$)9$\"\"#F1#!\"\"F9*&#F1\"\")F1*$)F0F9F1F1F;F1F1*(F7F1F0F1F2F1F;F*F*
F*RF'F*F+F*,&*&F8F1F2F1F1*&#F1\"\"%F1*(F8F1F@F1F2F1F1F;F*F*F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqs:=equate(delf(x,y),[0,0])
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqsG<$/,&*&%\"yG\"\"\"-%$expG6
#,&*$)%\"xG\"\"#F*#!\"\"F2*&#F*\"\")F**$)F)F2F*F*F4F*F**(F0F*F)F*F+F*F
4\"\"!/,&*&F1F*F+F*F**&#F*\"\"%F**(F1F*F9F*F+F*F*F4F;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "critpts:=solve(eqs,\{x,y\});" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(critptsG6'<$/%\"xG\"\"!/%\"yGF)<$/F
(\"\"\"/F+\"\"#<$F-/F+!\"#<$F//F(!\"\"<$F5F2" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 44 "We see there are 5 critical points given by:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "p1:=subs(critpts[1],[x,y]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G7$\"\"!F&" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "p2:=subs(critpts[2],[x,y]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#p2G7$\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 "p3:=subs(critpts[3],[x,y]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p3G7$\"\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "p4:=subs(critpts[4],[x,y]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p4G7$!\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "p5:=subs(critpts[5],[x,y]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p5G7$!\"\"!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 210 "Use
 the second derivative test to determine if each critical point is a m
aximum, a minimum or a saddle point.  Note, the test may fail.  First,
 compute the Hessian and the leading principal minor determinants: " }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Hf:=HESS(f):   matrix(Hf(x,
y));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,&*(%\"xG\"\"
\"%\"yGF+-%$expG6#,&*$)F*\"\"#F+#!\"\"F3*&#F+\"\")F+*$)F,F3F+F+F5F+!\"
$*()F*\"\"$F+F,F+F-F+F+,*F-F+*&F2F+F-F+F5*&#F+\"\"%F+*&F:F+F-F+F+F5**#
F+FCF+F2F+F:F+F-F+F+7$F?,&F)#F;FC**#F+\"#;F+F*F+)F,F>F+F-F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "leading_principal_minor_dete
rminants(Hf(x,y)):" }}{PARA 6 "" 1 "" {TEXT -1 37 "Leading Principal M
inor Determinants:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"\",
&*(%\"xGF'%\"yGF'-%$expG6#,&*$)F*\"\"#F'#!\"\"F2*&#F'\"\")F'*$)F+F2F'F
'F4F'!\"$*()F*\"\"$F'F+F'F,F'F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%
\"DG6#\"\"#,2*()%\"xGF'\"\"\")%\"yGF'F,)-%$expG6#,&*$F*F,#!\"\"F'*&#F,
\"\")F,*$F-F,F,F6F'F,#\"\"&\"\"%*&#F,\"#;F,*(F*F,)F.F=F,F/F,F,F6*&#F,F
=F,*()F+F=F,F-F,F/F,F,F6*$F/F,F6*(F'F,F*F,F/F,F,*(#F,F'F,F-F,F/F,F,*&F
FF,F/F,F6*&#F,F@F,*&FBF,F/F,F,F6" }}}{PARA 0 "" 0 "" {TEXT -1 116 "At \+
each critical point, evaluate the Hessian and the leading principal mi
nor determinants and interpret the results:" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 "'p1'=p1;   H1:=Hf(op(p1));   LPMD(H1):" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/%#p1G7$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#H1G7$7$\"\"!\"\"\"7$F(F'" }}{PARA 6 "" 1 "" {TEXT -1 37 "Lead
ing Principal Minor Determinants:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
&%\"DG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"#!
\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 73 "Since D[2] is negative, the firs
t critical point (0,0) is a saddle point." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "'p2'=p2;   H2:=Hf(op(p2));   LPMD(H2):" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/%#p2G7$\"\"\"\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#H2G7$7$,$-%$expG6#!\"\"!\"%\"\"!7$F-,$F(F+" }}{PARA 
6 "" 1 "" {TEXT -1 37 "Leading Principal Minor Determinants:" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"\",$-%$expG6#!\"\"!\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"#,$*$)-%$expG6#!\"\"F'\"
\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 96 "Since D[2] is positive and \+
D[1] is negative, the second critical point (1,2) is a local maximum.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'p3'=p3;   H3:=Hf(op(p3)
);   LPMD(H3):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#p3G7$\"\"\"!\"#" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H3G7$7$,$-%$expG6#!\"\"\"\"%\"\"!
7$F-F(" }}{PARA 6 "" 1 "" {TEXT -1 37 "Leading Principal Minor Determi
nants:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"\",$-%$expG6#!
\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"#,$*$)-%$exp
G6#!\"\"F'\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 96 "Since D[2] is p
ositive and D[1] is positive, the third critical point (1,-2) is a loc
al minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'p4'=p4;   H4
:=Hf(op(p4));   LPMD(H4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#p4G7$!
\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H4G7$7$,$-%$expG6#!\"
\"\"\"%\"\"!7$F-F(" }}{PARA 6 "" 1 "" {TEXT -1 37 "Leading Principal M
inor Determinants:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"\",
$-%$expG6#!\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"DG6#\"\"#
,$*$)-%$expG6#!\"\"F'\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 100 "Sin
ce D[2] is positive and D[1] is positive, the fourth critical point   \+
 (-1,2) is a local minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "'p5'=p5;   H5:=Hf(op(p5));   LPMD(H5):" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%#p5G7$!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#H5G7$7$,$-%$expG6#!\"\"!\"%\"\"!7$F-,$F(F+" }}{PARA 6 "" 1 "" {TEXT 
-1 37 "Leading Principal Minor Determinants:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"DG6#\"\"\",$-%$expG6#!\"\"!\"%" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/&%\"DG6#\"\"#,$*$)-%$expG6#!\"\"F'\"\"\"\"\"%" }}}
{PARA 0 "" 0 "" {TEXT -1 100 "Since D[2] is positive and D[1] is negat
ive, the fifth critical point    (-1,-2) is a local maximum." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "To confirm the \+
conclusions, use a contour plot:    (Try rotating the plot.)" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "contourplot3d(f(x,y), x=-2..
2, y=-3..3, axes=boxed, thickness=2, numpoints=2000, shading=zhue); " 
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4803FC7C152FB3CE2593FCB55E5FE7A7F663FCE9328B4C50F7D3FD0B875407F897E3FD
1F474D48F5FF63FD2FB727527B2953FD3CC4F863A8E493FD466EA72C5BB7F3FD4CC135
870EDED3FD4FD7B01F1CF6D3FD4FD9CFF12DC943FD4CFA5C9119F313FD47755F6E0934
53FD3F8E38E6FE2D63FD358DA7EC077103FD29BFD42F7DBF03FD1C7269A268B183FD0D
F2D237FA36C-%'COLOURG6#%%ZHUEG-%&STYLEG6#%(CONTOURG-%+AXESLABELSG6%%\"
xG%\"yGQ!6\"-%*THICKNESSG6#F+-%*AXESSTYLEG6#%$BOXG-%+PROJECTIONG6%$!#!
*F)$F)F)\"\"\"" 1 7 0 1 10 2 2 4 1 2 2 1.000000 0.000000 -90.000000 1 
0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 106 "(1,2) and (-1,-2) are a \+
local maxima.  (1,-2) and (-1,2) are a local minima.  And (0,0) is a s
addle point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 24 "The Constrained Problem:" }}{PARA 0 "" 
0 "" {TEXT -1 97 "Find the absolute maximum and minimum values of a fu
nction inside or on the boundary of a region." }}{SECT 0 {PARA 4 "" 0 
"" {TEXT 269 8 "Example:" }}{PARA 0 "" 0 "" {TEXT -1 23 "Extremize the
 function:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:=(x,y)->x*y*
exp(-x^2/2-y^2/8); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%
\"yG6\"6$%)operatorG%&arrowGF)*(9$\"\"\"9%F/-%$expG6#,&*$)F.\"\"#F/#!
\"\"F7*&#F/\"\")F/*$)F0F7F/F/F9F/F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 
39 "inside or on the ellipse g(x,y)=1 where" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "g:=(x,y)->x^2/4 + y^2/16;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gGR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*$)9$\"
\"#\"\"\"#F2\"\"%*&#F2\"#;F2)9%F1F2F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 270 9 "Solution
:" }}{PARA 0 "" 0 "" {TEXT -1 142 "The interior critical points were f
ound in the unconstrained example.  There are three methods of finding
 the critical points on the boundary." }}{SECT 0 {PARA 5 "" 0 "" 
{TEXT -1 42 "Boundary Method I:   Eliminate a Variable " }}{PARA 0 "" 
0 "" {TEXT -1 38 "Solve the constraint for one variable:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "y0:=solve(g(x,y)=1,y);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#y0G6$,$*$-%%sqrtG6#,&*$)%\"xG\"\"#\"\"\"!\"
\"\"\"%F0F0F/,$F'!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 212 "Notice that we
 named the solution y0 instead of y so that we can still use y as a va
riable.  Also notice that there are 2 solutions for the upper and lowe
r halves of the ellipse.  We must handle these separately." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "For the upper hal
f of the boundary, substitute the solution into the function and diffe
rentiate:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f1:=makefunctio
n(x,f(x,y0[1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1GR6#%\"xG6\"6
$%)operatorG%&arrowGF(,$*(9$\"\"\"-%%sqrtG6#,&*$)F.\"\"#F/!\"\"\"\"%F/
F/-%$expG6#!\"#F/F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "Df1:=D(f1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Df1GR6#%\"xG6\"6
$%)operatorG%&arrowGF(,&*&-%%sqrtG6#,&*$)9$\"\"#\"\"\"!\"\"\"\"%F6F6-%
$expG6#!\"#F6F5*&*(F5F6F3F6F9F6F6*$-F/6#F1F6F7F7F(F(F(" }}}{PARA 0 "" 
0 "" {TEXT -1 90 "Find the x-coordinate at each critical point and sub
stitute back to find the y-coordinate:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "x0:=solve(Df1(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#x0G6$*$-%%sqrtG6#\"\"#\"\"\",$F&!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 40 "y1:=subs(x=x0[1],y0[1]); b1:=[x0[1],y1];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G,$*$-%%sqrtG6#\"\"#\"\"\"F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G7$*$-%%sqrtG6#\"\"#\"\"\",$F&F*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y2:=subs(x=x0[2],y0[1])
; b2:=[x0[2],y2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G,$*$-%%sqrt
G6#\"\"#\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G7$,$*$-%%sqr
tG6#\"\"#\"\"\"!\"\",$F'F+" }}}{PARA 0 "" 0 "" {TEXT -1 42 "Repeat for
 the lower half of the boundary:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 31 "f2:=makefunction(x,f(x,y0[2]));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#f2GR6#%\"xG6\"6$%)operatorG%&arrowGF(,$*(9$\"\"\"-%%sqrtG6#,&
*$)F.\"\"#F/!\"\"\"\"%F/F/-%$expG6#!\"#F/F<F(F(F(" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 11 "Df2:=D(f2);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$Df2GR6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&-%%sqrtG6#,&*$)9$\"\"#
\"\"\"!\"\"\"\"%F6F6-%$expG6#!\"#F6F<*&*(F5F6F3F6F9F6F6*$-F/6#F1F6F7F6
F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x0:=solve(Df2(x)=
0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G6$*$-%%sqrtG6#\"\"#\"\"
\",$F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y3:=subs(x=x
0[1],y0[2]); b3:=[x0[1],y3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G
,$*$-%%sqrtG6#\"\"#\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b3
G7$*$-%%sqrtG6#\"\"#\"\"\",$F&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "y4:=subs(x=x0[2],y0[2]); b4:=[x0[2],y4];" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#y4G,$*$-%%sqrtG6#\"\"#\"\"\"!\"#" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#b4G7$,$*$-%%sqrtG6#\"\"#\"\"\"!\"\",$F'!
\"#" }}}{PARA 0 "" 0 "" {TEXT -1 139 "Finally, we tabulate the values \+
of the function at all interior and boundary critical points and ident
ify the absolute maximum and minimum:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "p1; f(op(p1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$
\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 24 "p2; f(op(p2)); evalf(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$e
xpG6#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+C))edt!#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "p3; f(op(p3)); evalf(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$-%$expG6#!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#$!+C))edt!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "p4; f(op(p
4)); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$!\"\"\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"\"!\"#" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$!+C))edt!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "p5; f(op(p5)); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$
!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"\"\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+C))edt!#5" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "b1; f(op(b1)); evalf(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$*$-%%sqrtG6#\"\"#\"\"\",$F$F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$-%$expG6#!\"#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#$\"+G8T8a!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b2; f(op(b
2)); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*$-%%sqrtG6#\"\"
#\"\"\"!\"\",$F%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"#!
\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+G8T8a!#5" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 24 "b3; f(op(b3)); evalf(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7$*$-%%sqrtG6#\"\"#\"\"\",$F$!\"#" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$-%$expG6#!\"#!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#$!+G8T8a!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b4; f(op(b
4)); evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$*$-%%sqrtG6#\"\"
#\"\"\"!\"\",$F%!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#!\"
#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G8T8a!#5" }}}{PARA 0 "" 
0 "" {TEXT -1 154 "So we see that the absolute maxima occur at the int
erior points (1,2) and (-1,-2), and the absolute minima occur at the i
nterior points (1,-2) and (-1,2)." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 47 "Boundary Me
thod II:   Parametrize the Boundary " }}{PARA 0 "" 0 "" {TEXT -1 27 "D
efine the parametrization:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
39 "r:=makefunction(t,[2*cos(t),4*sin(t)]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rG7$R6#%\"tG6\"6$%)operatorG%&arrowGF),$-%$cosG6#9$
\"\"#F)F)F)RF'F)F*F),$-%$sinGF0\"\"%F)F)F)" }}}{PARA 0 "" 0 "" {TEXT 
-1 38 "Restrict the function to the boundary:" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 42 "fr:=makefunction(t,simplify(f(op(r(t)))));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#frGR6#%\"tG6\"6$%)operatorG%&arrowG
F(,$*(-%$cosG6#9$\"\"\"-%$sinGF0F2-%$expG6#!\"#F2\"\")F(F(F(" }}}
{PARA 0 "" 0 "" {TEXT -1 41 "Find the critical points on the boundary:
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Dfr:=D(fr);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$DfrGR6#%\"tG6\"6$%)operatorG%&arrowGF(,&*&)-
%$sinG6#9$\"\"#\"\"\"-%$expG6#!\"#F4!\")*(\"\")F4)-%$cosGF1F3F4F5F4F4F
(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "bndcritpts:=solve(
Dfr(t)=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+bndcritptsG6$,$%#Pi
G#!\"\"\"\"%,$F'#\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "b1:=r(bndcritpts[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G7$*
$-%%sqrtG6#\"\"#\"\"\",$F&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "b2:=r(bndcritpts[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2
G7$*$-%%sqrtG6#\"\"#\"\"\",$F&F*" }}}{PARA 0 "" 0 "" {TEXT -1 112 "Sin
ce the equation is non-polynomial, solve may not give all solutions.  \+
So we plot the function for one period:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "plot(Dfr,-Pi..Pi);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 
300 300 {PLOTDATA 2 "6%-%'CURVESG6$7]s7$$!3*****4tk#fTJ!#<$\"3i,H*eE#o
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#yo#FU7$$\"31?#=([fA^DF*$\"3t.>!)zD'o6%FU7$$\"3A?c*)yu\"3i#F*$\"335<*o
guiY&FU7$$\"35TH4t41$o#F*$\"3aD,57[P%e'FU7$$\"3-i-HnWIXFF*$\"3W&HJ*)fo
0g(FU7$$\"3Q!4nV,p;\"GF*$\"3!*Q%*ou%>Tb)FU7$$\"3u=RWhN.yGF*$\"3Y'pO)GM
>d$*FU7$$\"3tX=#4!HbTHF*$\"3'G.F6!)G=(**FU7$$\"3ss(*RSA20IF*$\"34+Tv)o
uD/\"F*7$$\"3;HB\"HM-#RIF*$\"3Kt]R!Hg+1\"F*7$$\"3/')[UXCLtIF*$\"3[YXw0
&3E2\"F*7$$\"3[k6o'\\(R!4$F*$\"3eD/+B>,x5F*7$$\"3#HWPzaiu5$F*$\"3%)3\"
4(*)3;!3\"F*7$$\"3O@P>*fFX7$F*$\"3m?!H\"Q<0#3\"F*7$$\"3!)***\\/l#fTJF*
F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F`dmF_dm-%+AXESLABELSG6$Q!6\"Fddm-
%%VIEWG6$;$!+aEfTJ!\"*$\"+aEfTJF\\em%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 
2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" 
{TEXT -1 85 "From the plot and its symmetries, it is obvious that solv
e missed two more solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "b3:=r(3*Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b3G7$,$*$-%%s
qrtG6#\"\"#\"\"\"!\"\",$F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "b4:=r(-3*Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b4G7$,$*$-
%%sqrtG6#\"\"#\"\"\"!\"\",$F'!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 206 "Fi
nally, we tabulate the values of the function at all interior and boun
dary critical points and identify the absolute maximum and minimum: \n
(This was done with the first method.  So we won't redo it here.)" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 0 "
" {TEXT -1 49 "Boundary Method III:   Lagrange Multiplier Method" }}
{PARA 0 "" 0 "" {TEXT -1 69 "Find the gradient of the function and the
 gradient of the constraint:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "delf:=GRAD(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%delfG7$R6$%\"
xG%\"yG6\"6$%)operatorG%&arrowGF*,&*&9%\"\"\"-%$expG6#,&*$)9$\"\"#F1#!
\"\"F9*&#F1\"\")F1*$)F0F9F1F1F;F1F1*(F7F1F0F1F2F1F;F*F*F*RF'F*F+F*,&*&
F8F1F2F1F1*&#F1\"\"%F1*(F8F1F@F1F2F1F1F;F*F*F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "delg:=GRAD(g);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%delgG7$R6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF*,$9$#\"\"\"\"\"
#F*F*F*RF'F*F+F*,$9%#F1\"\")F*F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 91 "Se
t up the Lagrange multiplier equations and solve for the critical poin
ts on the boundary:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqs:=
equate(delf(x,y),lambda*delg(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%$eqsG<$/,&*&%\"xG\"\"\"-%$expG6#,&*$)F)\"\"#F*#!\"\"F1*&#F*\"\")F**
$)%\"yGF1F*F*F3F*F**&#F*\"\"%F**(F)F*F8F*F+F*F*F3,$*&%'lambdaGF*F9F*#F
*F6/,&*&F9F*F+F*F**(F0F*F9F*F+F*F3,$*&F@F*F)F*#F*F1" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 44 "sol:=solve(\{op(eqs),g(x,y)=1\},\{x,y,lam
bda\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG6$<%/%\"yG,$-%'RootO
fG6$,&*$)%#_ZG\"\"#\"\"\"F2F1!\"\"/%&labelG%$_L8GF1/%\"xGF*/%'lambdaG,
$-%$expG6#!\"#!\"%<%F'/F8,$F*F3/F:,$F<\"\"%" }}}{PARA 0 "" 0 "" {TEXT 
-1 70 "Since the solutions involve a RootOf, we resolve them using all
values:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sol1:=allvalues(s
ol[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G6$<%/%\"xG*$-%%sqrt
G6#\"\"#\"\"\"/%\"yG,$F)F-/%'lambdaG,$-%$expG6#!\"#!\"%<%/F(,$F)!\"\"/
F0,$F)F8F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b1:=subs(sol1
[1],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G7$*$-%%sqrtG6#\"
\"#\"\"\",$F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b4:=subs
(sol1[2],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b4G7$,$*$-%%sqr
tG6#\"\"#\"\"\"!\"\",$F'!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "sol2:=allvalues(sol[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%s
ol2G6$<%/%\"xG,$*$-%%sqrtG6#\"\"#\"\"\"!\"\"/%\"yG,$F*F./%'lambdaG,$-%
$expG6#!\"#\"\"%<%/F(F*/F2,$F*F:F4" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "b3:=subs(sol2[1],[x,y]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#b3G7$,$*$-%%sqrtG6#\"\"#\"\"\"!\"\",$F'F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b2:=subs(sol2[2],[x,y]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G7$*$-%%sqrtG6#\"\"#\"\"\",$F&!\"
#" }}}{PARA 0 "" 0 "" {TEXT -1 208 "Finally, we tabulate the values of
 the function at all interior and boundary critical points and identif
y the absolute maximum and minimum: \n  (This was done with the first \+
method.  So we won't redo it here.)" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{MARK "15 3 28 1 0" 2 }{VIEWOPTS 1 1 
0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }