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1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 20 "Lagrange Multipliers" }} {PARA 19 "" 0 "" {TEXT -1 14 "Brian E. Blank" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 " Constrained Extrema" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Consider the path " }{XPPEDIT 18 0 " proc (t) options operator, arrow; [1-t, 2-t/2, 10-(1-t)^2-2*(2-t/2)^2] end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"7%,&\"\"\"F-F%!\"\" ,&\"\"#F-*&F%F-F0F.F.,(\"#5F-*$,&F-F-F%F.F0F.*&F0F-*$,&F0F-*&F%F-F0F.F .F0F-F.F*F*F*" }{TEXT -1 23 " \n \non the surface " }{XPPEDIT 18 0 "z = 10-x^2-2*y^2;" "6#/%\"zG,(\"#5\"\"\"*$%\"xG\"\"#!\"\"*&F*F'*$% \"yGF*F'F+" }{TEXT -1 181 ". \n\nThe surface may be imagined as a hil l. We may imagine that we are following a path over the hill. Relative to the ground our path follows the aquamarine line in the plot below. 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The point at which our path reaches a maximum height does n ot occur at the peak of the hill. It is not a critical point of the he ight function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "The maximum that we attain is a function not only of the particular surface but also on the path that we follow. We say that t he variables " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 112 " are constrained to lie o n this path. How do we use calculus to find an extremum associated wit h a constraint? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Let us examine the level curves of " }{XPPEDIT 18 0 "f; " "6#%\"fG" }{TEXT -1 3 ":\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "contourPlot := contourplot(f(x,y), x = -2.. 2, y= -2..2, contou rs=27,color=COLOR(RGB,0.75,0.7,0.7)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "projectionPlot := plot([1-t, 2-t/2, t=0..3], color=aq uamarine, thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " display(contourPlot,projectionPlot);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6@-%'CURVESG6U7$7$$!3w-++++++c!#=$!3l%*\\P4r& Ga#F*7$$!3%p*\\P4r&Gu&F*$!3_.++++++CF*7$F-7$$!3!y)*\\(=Ur&)eF*$!3w=+D \"y&G9@F*7$7$$!3a'*\\P4r&Ga'F*$!3)[.++++++)!#>F37$F97$F:$\"3U)))\\P4r& G9F>7$7$F:$\"3[l************zF>F@7$FD7$$!3h)**\\iSr&GiF*$\"3d\"**\\iSr &G9F*7$7$F.$\"3e'************R#F*FH7$FN7$$!3%)*\\(oa&G9n&F*$\"3c#\\(oa &G9Z#F*7$7$F($\"3V()\\P4r&Ga#F*FR7$F'7$$!3!\\.+Dcr&G]F*$!3ws**\\P%G9(H 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"Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26 " "Curve 27" "Curve 28" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 76 "As we ascend, passing from one height to another, \+ our path relative to the " }{XPPEDIT 18 0 "xy;" "6#%#xyG" }{TEXT -1 12 "-plane the " }{TEXT 290 7 "crosses" }{TEXT -1 18 " level curves \+ of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 173 ". At the point at wh ich we reach our maximum we no longer pass to a higher level. Our path begines to cross the level curves we have previously crossed. This su ggests that " }{TEXT 291 94 "at a constrained extremum the constraint curve should be tangent to the level curve it touches" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Suppo se that the constraint is expressed as " }{XPPEDIT 18 0 "phi(x,y) = co nstant;" "6#/-%$phiG6$%\"xG%\"yG%)constantG" }{TEXT -1 24 ". Then the \+ gradient of " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 36 " is per pendicular to the graph of " }{XPPEDIT 18 0 "phi(x,y) = constant;" "6 #/-%$phiG6$%\"xG%\"yG%)constantG" }{TEXT -1 22 " . The gradient of \+ " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 43 " is perpendicular to th e level curves of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 99 " . Th erefore the analytic condition that expresses the tangency at a constr ained extremum is that " }{TEXT 292 15 "the gradient of" }{TEXT -1 3 " " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 3 " " }{TEXT 293 20 "i s a scalar multiple" }{TEXT -1 2 " " }{XPPEDIT 18 0 "lambda;" "6#%'la mbdaG" }{TEXT -1 2 " " }{TEXT 294 22 " times the gradient of" }{TEXT -1 2 " " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The scalar mult iplier " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 16 " is cal led a " }{TEXT 295 19 "Lagrange multiplier" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 " Lagrange Multiplier Problems\n" }}{PARA 0 "" 0 "" {TEXT -1 131 "Let us illustrate these ideas by working thro ugh a classic constrained extremum problem. It is traditionally referr ed to as \nthe " }{TEXT 259 16 "Milkmaid Problem" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 8 "Example: " }}{PARA 0 "" 0 "" {TEXT 261 73 "Suppose that a farmhand lives in a h ouse that is represented by the point" }{TEXT -1 1 " " }{TEXT 260 9 "P = [0,0]" }{TEXT -1 2 ". " }{TEXT 262 40 "Suppose that the farmhand ha s to go from" }{TEXT -1 2 " " }{TEXT 263 1 "P" }{TEXT -1 2 " " } {TEXT 267 20 "to a barn located at" }{TEXT -1 1 " " }{TEXT 264 10 "Q = [10,0]" }{TEXT -1 83 ". However, the farmhand must first stop at the \+ river, represented by the equation " }{XPPEDIT 18 0 "x/16-y/5 = 1;" " 6#/,&*&%\"xG\"\"\"\"#;!\"\"F'*&%\"yGF'\"\"&F)F)F'" }{TEXT -1 2 ", " } {TEXT 269 80 "in order to fill a bucket of water to bring to the barn \+ animals. \n\nAt what point" }{TEXT -1 1 " " }{TEXT 265 1 "R" }{TEXT -1 1 " " }{TEXT 268 69 "on the river should the farmhand go in order t o minimize the distance" }{TEXT -1 1 " " }{TEXT 266 5 "PR+RQ" }{TEXT -1 2 "? 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