"3m3'yp;:g#[F-$!3] *)ppW,)Gw)F-7$$\"3YyTOS#=A$[F-$!3/,JN)zD_I)F-7$$\"3#RJEJ.D\"Q[F-$!3!H+ -1877\"zF-7$$\"3l].r^\")GW[F-$!3oqLD0c[F-$!3!))p/b\"Qv7pF-7$$\"3$fI5rh$Ri[F-$!3b+q&F-7$$\"3]&)*=,Y*)=*[F-$!3E!e`&QL06bF-7$$\"3gu39FDO)*[F-$!30\"3)HPU (\\J&F-7$$\"3LG8wl)yT!\\F-$!3/i8)z,_*\\^F-7$$\"3Y#G6([!*Q5\\F-$!3$G>60 9IV)\\F-7$$\"3_8'yuR;g\"\\F-$!3/F:)\\'e'G%[F-7$$\"3z&*)egdo@#\\F-$!3'Q )3;&Hvnp%F-7$$\"3'HX_mHdz#\\F-$!3mp]iM4#oc%F-7$$\"3/Vnhx\"3S$\\F-$!3G4 Yn%RG\"QWF-7$$\"3J#of(>S#*R\\F-$!3y6::/%>)=VF-7$$\"3!\\iJ#et6Y\\F-$!3, Zd[XfF+UF-7$$\"3KUeFzA3_\\F-$!3%f+(pxEz\"4%F-7$$\"3q\">wDF#=e\\F-$!3SJ c[#G'>')RF-7$$\"3/$>_[vJU'\\F-$!3\\S^8WmU')QF-7$$\"3L9*)y+1zp\\F-$!3_ \")**4'\\(z)z$F-7$$\"3MRyal;;w\\F-$!3(yq-,M3Gq$F-7$$\"3SN#==-g=)\\F-$! 3)4mW-N72i$F-7$$\"37z67ec$z)\\F-$!3ewX0()>$o`$F-7$$\"3!*>oH<3v$*\\F-$! 3?[hoii&)fMF-7$$\"\"&F*$!3%eeYi+:0Q$F--%'COLOURG6&%$RGBG$\"*++++\"!\") $F*F*F_bq-%+AXESLABELSG6$Q\"x6\"Q\"yFdbq-%%VIEWG6$;F(Fdaq;$F]^lF*$\"#5 F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "An Example of Finding the Inf imum Analytically" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "As an example , we will prove that the infimum of " }{XPPEDIT 18 0 "cosh(x)" "6#-%%c oshG6#%\"xG" }{TEXT -1 23 " on the real line is 1." }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "cosh(0) = 1" "6#/-%%coshG6#\"\"!\"\"\"" }{TEXT -1 52 " \+ and therefore 1 is a lower bound. Differentiating " }{XPPEDIT 18 0 "c osh(x)" "6#-%%coshG6#%\"xG" }{TEXT -1 17 " with respect to " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 8 ", we get" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "diff( cosh(x), x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%sinhG6#%\"xG" }}}{EXCHG {PARA 257 "" 1 "" {TEXT -1 60 "which is a function which is strictly greater than zero for " } {XPPEDIT 18 0 "x > 0" "6#2\"\"!%\"xG" }{TEXT -1 33 " and strictly less than zero for " }{XPPEDIT 18 0 "x < 0" "6#2%\"xG\"\"!" }{TEXT -1 43 " . By the mean value theorem, we have that " }{XPPEDIT 18 0 "cosh(x) > cosh(0)" "6#2-%%coshG6#\"\"!-F%6#%\"xG" }{TEXT -1 9 " for all " } {XPPEDIT 18 0 "x <> 0" "6#0%\"xG\"\"!" }{TEXT -1 9 ". Since " } {XPPEDIT 18 0 "cosh(x) " "6#-%%coshG6#%\"xG" }{TEXT -1 60 " is contino usly differentiable, using the mean value theorem" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 34 "Proof Using the Mean Value Theorem" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The mean value theorem states that for tw o values x and y where " }{XPPEDIT 18 0 "x < y" "6#2%\"xG%\"yG" } {TEXT -1 77 ", there exists some value c in the open interval (x,y) su ch that\n " }{XPPEDIT 18 0 "f(y)-f(x) = eval(diff(f(x),x),x = c)*(y-x)" "6#/,&-%\"fG6#%\"yG\"\"\"-F&6#%\"xG!\"\"*&-%%evalG6$-%%di ffG6$-F&6#F,F,/F,%\"cGF),&F(F)F,F-F)" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 50 "In this case, if we assume y>0 this gives us that " } {XPPEDIT 18 0 "cosh(y) - cosh(0) = sinh(c)*y" "6#/,&-%%coshG6#%\"yG\" \"\"-F&6#\"\"!!\"\"*&-%%sinhG6#%\"cGF)F(F)" }{TEXT -1 70 " for some va lue c in the open interval (0,y). Therefore c>0 and thus " }{XPPEDIT 18 0 "0 < sinh(c);" "6#2\"\"!-%%sinhG6#%\"cG" }{TEXT -1 18 " and so th erefore " }{XPPEDIT 18 0 "0 < cosh(y)-1;" "6#2\"\"!,&-%%coshG6#%\"yG\" \"\"F*!\"\"" }{TEXT -1 52 ". A similar argument gives us the same res ult when " }{XPPEDIT 18 0 "y < 0" "6#2%\"yG\"\"!" }{TEXT -1 28 ". The refore the infimum of " }{XPPEDIT 18 0 "cosh(x) " "6#-%%coshG6#%\"xG" }{TEXT -1 6 " is 1." }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "we have t hat " }{XPPEDIT 18 0 "cosh(x) > cosh(0)" "6#2-%%coshG6#\"\"!-F%6#%\"xG " }{TEXT -1 80 " for all x other than zero, and so therefore 0 is a lo wer bound of the function " }{XPPEDIT 18 0 "cosh(x)" "6#-%%coshG6#%\"x G" }{TEXT -1 87 ". The value 1 must also be the greatest upper bound, for if there was any upper bound " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "beta > 1" "6#2\"\"\"%%betaG " }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "beta > cosh(0)" "6#2-%%coshG6# \"\"!%%betaG" }{TEXT -1 47 " and so therefore beta is not a lower boun d of " }{XPPEDIT 18 0 "cosh(x)" "6#-%%coshG6#%\"xG" }{TEXT -1 29 ". T herefore, the infimum of " }{XPPEDIT 18 0 "cosh(x)" "6#-%%coshG6#%\"xG " }{TEXT -1 21 " on the real line is " }{XPPEDIT 18 0 "1" "6#\"\"\"" } {TEXT -1 26 ", as is returned by Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "minimize( cosh(x), x=-infinity..infinity );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "The user may wish to try this technique with a simpler function su ch as " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 1 "." } {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 "Specifying the Domain of a Variable" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The domain of a variable can be specified in one of three ways:" }{MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "The Default " }{TEXT 259 19 "minimize( f(x), x )" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "If just the variable is listed, t hen it is assumed that that variable is to be minimized over the entir e real line, that is, " }{XPPEDIT 18 0 "-infinity..infinity" "6#;,$%)i nfinityG!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " minimize( sin(x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "A Range " }{TEXT 260 26 "minimize( f(x), x = 0..1 )" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "If the varia ble is equated to a range, then the variable is minimized over the clo sed interval bounded by the given range. The endpoints of the range m ust be of type " }{HYPERLNK 17 "realcons" 2 "type[realcons]" "" } {TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "minimize( sin(x), \+ x=0..4 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#\"\"%" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "This form of specifying the domai n over which the function is to be minimized allows a quick substituti on to give a plot of the function:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "minimize( x^2-x*y+2*y^2+exp(x), x=-1..0, y=-1/2..1/2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)-%)LambertWG6##\"\"%\"\"(\"\"#\"\"\"#F+ \"\")-%$expG6#,$F&!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+AFQ, \")!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "plot3d( x^2-x*y+2 *y^2+exp(x), x=-1..0, y=-1/2..1/2, 'axes=framed' );" }{TEXT -1 0 "" }} {PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6%-%%GRIDG6%;$!\" 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Some examples are:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "minimize(cosh(x), x=5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%coshG6#\"\"&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "minimize(x^2, x>5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "mini mize(sin(x), x<3 ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "minimize(signum(x), x>0);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "minimize(signum(x), x>=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}}{PARA 0 "" 0 "" {TEXT -1 142 "If no domains are given for any variable then it is assumed that each indeterminant of type name is being minimized over the entire real line:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "minimize( x^2+cosh(y)+3 );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "minimize( x^2+cosh(y)+3, x=-infinity..infinity, y=-in 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