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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 15 "Critical Points" }}{PARA 259 "
" 0 "" {TEXT -1 34 "We analyze the behavior of f(x,y)=" }{XPPEDIT 18 
0 "x^3 + y^3 + 3*x*y + 3" ",**$%\"xG\"\"$\"\"\"*$%\"yG\"\"$F&*(\"\"$F&
F$F&F(F&F&\"\"$F&" }{TEXT -1 25 " near its critical points" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 260 "" 0 "" {TEXT -1 33 "We first
 find the critical points" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "resta
rt;\nf := x^3 + y^3 + 3*x*y + 3;\nfx := diff(f,x);\nfy := diff(f,y);\n
critpts := solve( \{fx=0, fy=0\}, \{x,y\} );\nallvalues( critpts[3] );
" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 126 "So there are two critical \+
points (0,0) and (-1,-1). We examine the behavior of f(x,y) near them.
\nPlotting f(x,y) near (-1,-1) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 
"plot3d( x^3 + y^3 + 3*x*y + 3, x=-1.5..-0.5, y=-1.5..-0.5, axes=boxed
,\n        style=patchcontour, orientation=[36,82] );" }}}{EXCHG 
{PARA 258 "" 0 "" {TEXT -1 37 "So f(x,y) has a local max at (-1,-1)." 
}}{PARA 261 "" 0 "" {TEXT -1 73 "Now we examine the other critical poi
nt (0,0). Plotting f(x,y) near (0,0)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 120 "plot3d( x^3 + y^3 + 3*x*y + 3, x=-0.4..0.3, y=-0.4..0.3, axes=b
oxed,\n        style=patchcontour, orientation=[160,75] );" }}}{EXCHG 
{PARA 262 "" 0 "" {TEXT -1 69 "So (0,0) is neither a local max nor a l
ocal min for f(x,y) - it is a " }{TEXT 259 12 "saddle point" }{TEXT 
-1 1 "." }}{PARA 263 "" 0 "" {TEXT 260 69 "Conclusion : A critical poi
nt need not be a local max or a local min." }}}}{MARK "2 0" 0 }
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