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YLEG6#%-PATCHCONTOURG-%*AXESSTYLEG6#%$BOXG" 1 4 0 1 10 0 2 4 1 2 2 1.000000 0.000000 70.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 54 "#Az \341br\341k alapj\341n 2 lok. min. \351s 2 lok. max. hely van." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "gx:=diff (g,x); gy:=diff(g,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gxG,&*&%\" yG\"\"\"-%$expG6#,(*$)%\"xG\"\"#F(!\"\"*&F/F(F'F(F(*&\"\"$F()F'F0F(F1F (F(**F/F(F'F(,&*&F0F(F/F(F1F'F(F(F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gyG,&*&%\"xG\"\"\"-%$expG6#,(*$)F'\"\"#F(!\"\"*&F'F(%\"yGF(F( *&\"\"$F()F2F/F(F0F(F(**F'F(F2F(,&F'F(*&\"\"'F(F2F(F0F(F)F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(\{gx,gy\},\{x,y\});" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"xG\"\"!/%\"yGF&<$/F(-%'RootOfG6 #,(\"\"\"F/*&\"#7F/)%#_ZG\"\"#F/!\"\"*&\"#LF/)F3\"\"%F/F//F%,&*&F7F/)F +\"\"$F/F/*&\"\"'F/F+F/F5" }}}{EXCHG }{EXCHG }{EXCHG }{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 60 "#az y \351rt\351kek:\nevalf(allvalues(RootOf (1-12*_Z^2+33*_Z^4))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$!+I<]S[!#5 $\"+I<]S[F%$!+\")HF'f$F%$\"+\")HF'f$F%" }}}{EXCHG }{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 93 "#az x \351rt\351kek:\nevalf(allvalues(33*RootO f(1-12*_Z^2+33*_Z^4)^3-6*RootOf(1-12*_Z^2+33*_Z^4))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$!*%\\*RQ)!\"*$\"*%\\*RQ)F%$\"*`F*GiF%$!*`F*GiF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "#Teh\341t 5 stac. pont va n a sz\341m\355t\341sok szerint, \366sszhangban a rajzzal. (A (0,0) ny eregpont.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "#2.b.)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "gxx:=diff(g,x,x): gxy:=diff( g,x,y): gyy:=diff(g,y,y):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "d1:=gxx: d2:=gxx*gyy-(gxy)^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "#Lok. max. az \341bra alapj\341n [0,2]X[0,2]-n:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "fsolve(\{gx,gy\},\{x,y\}, x= 0..2,y=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG$\"+I\\*RQ)!# 5/%\"yG$\"+H<]S[F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf (subs(%, [d1,d2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!+IU')4^!#5$ \"+#3A.h(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "#Teh\341t a \+ Hesse-m\341trix nagat\355v definit, \355gy a pont lok. max. hely." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "#Lok. max. az \341bra alapj \341n [0,2]X[-2,0]-n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "fs olve(\{gx,gy\},\{x,y\}, x=0..2,y=-2..0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG$\"+Dv#*Gi!#5/%\"yG$!+$)HF'f$F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(subs(%, [d1,d2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+437sP!#5$\"+'=d2?%F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 65 "#Teh\341t a Hesse-m\341trix pozit\355v definit, \+ \355gy a pont lok. min. hely." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "#Nyeregpont az \341bra alapj\341n (0,0)-ban:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "fsolve(\{gx,gy\},\{x,y\}, x=-0.1..0.1,y=-0. 1..0.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG$\"\"!F'/%\"yGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(subs(%, [d1,d2])); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"\"!F%$!\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "#Ez t\351nyleg indefinit, teh\341t \+ (0,0) nyeregpont." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "#3. feladat" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been red efined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A:=matrix(3,3, [1,2,3, 3,0,-1, 5,1,8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"\"\"#\"\"$7%F,\"\"!!\"\"7%\"\"&F*\" \")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "B:=matrix(2,3, [10,2 0,-3, 3,-2,7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7 $7%\"#5\"#?!\"$7%\"\"$!\"#\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "C:=matrix(3,1, [-1,3,8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"CG-%'matrixG6#7%7#!\"\"7#\"\"$7#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalm(B&*A+5*B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %'matrixG6#7$7%\"$0\"\"$<\"!#H7%\"#Z\"\"$\"$-\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "evalm(A&*C);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'matrixG6#7%7#\"#H7#!#67#\"#i" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalm(inverse(A));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%'matrixG6#7%7%#!\"\"\"#[#\"#8F*#\"\"\"\"#C7%#\"#HF*#\"\"(F*#!\"&F/7 %#F)\"#;#!\"$F9#F.\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " #A saj\341t\351rt\351kei, saj\341tvektorai ellen\365rz\351ssel:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ev:=eigenvectors(A):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "lambda1:=evalf(ev[1][1]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lambda1G^$$\"+I>PE(*!\"*$!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "lambda2:=evalf(ev[2][1]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lambda2G^$$!+ED<9E!\"*$\"\"$!#5 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "lambda3:=evalf(ev[3][1] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lambda3G^$$\"+'f+y)=!\"*$\"\" $!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "v1:=op(evalf(ev[1][ 3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G-%'vectorG6#7%$\"\"\"\" \"!^$$\")&en+\"!\"*$\"+X!yt;\"!#=^$$\"+>*y?!HF/$!#5F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "v2:=op(evalf(ev[2][3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G-%'vectorG6#7%$\"\"\"\"\"!^$$!+.-g\"G \"!\"*$\"+&eJWM%!#>^$$!+&RSK]$!#5$!#9!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "v3:=op(evalf(ev[3][3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v3G-%'vectorG6#7%$\"\"\"\"\"!^$$\"+g(*Q9A!\"*$\"+A68 H?!#>^$$!+4jK!=\"F/$F*!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "# Megjegyz\351s. L\341tszik, hogy az I-k kerek\355t\351si hib\341 k a komplex sz\341mokkal val\363\n# sz\341mol\341si hib\341k miatt. El t\374ntethet\365k az evalc-vel, pl.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalc(ev[3][1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, (*&\"#8#\"\"\"\"\"#-%$cosG6#,$*&#F'\"\"$F'-%'arctanG6#,$*(F(F'\"#@!\" \"\"$R%F&F'F'F'F'F6F/F'*(F/F&F%F&-%$sinGF+F'F'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "lambda3:=evalf(evalc(ev[3][1]));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%(lambda3G$\"+(f+y)=!\"*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "#Egy ellen\365rz\351s, pl. a m\341sodik p\341r os\351:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalf(evalm(A&*v 2-lambda2*v2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%^$$\" \"#!\"*$\"+qJ'))[\"!#>^$$\"\"!F0$\"+I%*=g;!#=^$$\"#