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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 41 "Max/Min of functions over bound
ed regions" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 101 "We explore where the largest and the smallest value of a func
tion are attained over a bounded region." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "From " }{TEXT 260 8 "Options " }
{TEXT -1 8 "choose  " }{TEXT 261 12 "Plot Display" }{TEXT -1 15 " then
 choose  W" }{TEXT 262 5 "indow" }{TEXT -1 1 "." }}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 20 "One dimensional case" }}{PARA 258 "" 0 "" {TEXT -1 
61 "find the largest and the smallest value of the function f(x)=" }
{XPPEDIT 18 0 "3*x^4 + 2*x^3 - 9*x^2" "6#,(*&\"\"$\"\"\"*$%\"xG\"\"%F&
F&*&\"\"#F&*$F(\"\"$F&F&*&\"\"*F&*$F(\"\"#F&!\"\"" }{TEXT 257 26 " ove
r the interval [-2,2]." }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 46 "We sta
rt by plotting the graph of the function" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "plot( 3*x^4 + 2*x^3 - 9*x^2, x=-2..2 );" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 262 "" 0 "" {TEXT -1 158 "From the
 graph it is clear that the largest value of f(x) is attained at the e
nd point x=2 and the smallest values is attained at the local min near
  x=-1.5. " }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" 
{TEXT -1 317 "In general, for functions of one variable, we know that \+
the largest/smallest value of a `smooth' function of one variable on a
n interval attained either at a local max/min inside the interval or a
t one of the end points (boundary) of the interval. We see if this ana
logy carries through to the multi-dimensional case." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 
"Two dimensional case" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
263 "" 0 "" {TEXT 263 86 "Find the largest and the smallest and the la
rgest value of the function (temperature)\n" }{XPPEDIT 264 0 "T(x,y) =
 (x-1/2)*(y-1/2)*exp(-(x^2+y^2)/4)" "6#/-%\"TG6$%\"xG%\"yG*(,&F'\"\"\"
*&\"\"\"F+\"\"#!\"\"F/F+,&F(F+*&\"\"\"F+\"\"#F/F/F+-%$expG6#,$*&,&*$F'
\"\"#F+*$F(\"\"#F+F+\"\"%F/F/F+" }{TEXT 265 1 "\n" }{TEXT -1 45 "over \+
the rectangular region x=-4..4, y=-4..4." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 54 "We start by plotting thi
s function over the rectangle." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "
T := (x-1/2)*(y-1/2)*exp( -(x^2+y^2)/4 );\nplot3d( T, x=-4..4, y=-4..4
, style=patchcontour, axes=boxed, \n                             shadi
ng=z, orientation=[55,65] );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 265 "" 0 "" {TEXT -1 416 "The graph of T(x,y) has two peaks and \+
two hollows. Largest value of T(x,y) occurs at one of the peaks and th
e min occurs at one of the hollows. \n\nTo obtain location of these po
ints - flatten picture from top (or change orientation = [-90,0]) in p
revious command and hit return. Remember red = high temperature and bl
ue = low temperature. \nMax is at (x,y) ~= (-1.2,-1.2)\nMin is at (x,y
) ~= (1.6,-1.2) and (-1.2, 1.2)\n" }}{PARA 266 "" 0 "" {TEXT -1 211 "C
onclusion  : The largest/smallest value seems to occur at one of the p
eaks/hollows of the function. However, from the one dimensional case w
e know that the answer is a little more complicated as we learn below.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 144 "We
 now find the largest and the smallest value of the same function T(x,
y) over the triangular plate bounded by the three lines x+y=0, x=4, y=
4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
258 53 "We start by plotting the function over this triangle." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "plot3d( T, x=-4..4, y=-4..-x, styl
e=patchcontour, axes=boxed, \n                              shading=z,
 orientation=[25,56] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 418 "T
he three dimensional picture is not as informative but you may see tha
t there is a single peak and no hollows. If you change the orientation
 to [-90,0] one has a colored plot of the function - remember red repr
esents high temperature and blue represents low temperature. You may n
otice that the highest temperature is still at a peak but the lowest t
emperature is not a hollow but on the boundary/edge of the plate.\n " 
}}{PARA 269 "" 0 "" {TEXT -1 129 "Conclusion : The largest/smallest va
lue of a \"smooth\" function over a region occurs either at a peak or \+
hollow inside the region " }{TEXT 259 44 "or at a point on the boundar
y of the region." }}}}{MARK "6" 0 }{VIEWOPTS 1 1 0 1 1 1803 }