(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 630306, 11280]*) (*NotebookOutlinePosition[ 630937, 11302]*) (* CellTagsIndexPosition[ 630893, 11298]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Example", "Subsubsection"], Cell[TextData[{ "Find the local maxima and minima of ", Cell[BoxData[ \(TraditionalForm\`f(x, y)\ = \ \(x\^2\) y\ - \ 2\ x\ y\ + \ 2 y\^2 - 15 y\)]], "." }], "Text"], Cell[TextData[{ "Solution.", StyleBox[" ", FontWeight->"Plain"] }], "Text", FontWeight->"Bold"], Cell[BoxData[ \(\(f[x_, y_]\ = \ \(x\^2\) y\ - \ 2\ x\ y\ + \ 2 y\^2 - 15 y;\)\)], "Input"], Cell["The first and second partial derivatives are:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x, y], x]\)], "Input"], Cell[BoxData[ \(\(-2\)\ y + 2\ x\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x, y], y]\)], "Input"], Cell[BoxData[ \(\(-15\) - 2\ x + x\^2 + 4\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x, y], {x, 2}]\)], "Input"], Cell[BoxData[ \(2\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x, y], {y, 2}]\)], "Input"], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x, y], x, y]\)], "Input"], Cell[BoxData[ \(\(-2\) + 2\ x\)], "Output"] }, Open ]], Cell["The crititcal points are:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cps\ = \ Solve[{D[f[x, y], x]\ \[Equal] \ 0, \ D[f[x, y], y] \[Equal] 0}]\)], "Input"], Cell[BoxData[ \({{y \[Rule] 0, x \[Rule] \(-3\)}, {y \[Rule] 0, x \[Rule] 5}, {y \[Rule] 4, x \[Rule] 1}}\)], "Output"] }, Open ]], Cell["\<\ Next we apply the Second Derivative Test at the criticial points.\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(d[x_, y_]\ = \ D[f[x, y], {x, 2}] D[f[x, y], {y, 2}] - D[f[x, y], x, y]^2\)], "Input"], Cell[BoxData[ \(\(-\((\(-2\) + 2\ x)\)\^2\) + 8\ y\)], "Output"] }, Open ]], Cell[TextData[{ "Plug the critical points into ", Cell[BoxData[ \(TraditionalForm\`d\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(d[x, y]\ /. \ cps\)], "Input"], Cell[BoxData[ \({\(-64\), \(-64\), 32}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x, y], {x, 2}]\ /. \ cps\)], "Input"], Cell[BoxData[ \({0, 0, 8}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f[x, y]\ /. \ cps\)], "Input"], Cell[BoxData[ \({0, 0, \(-32\)}\)], "Output"] }, Open ]], Cell[TextData[{ "Since ", Cell[BoxData[ \(TraditionalForm\`d(\(-3\), 0)\ = \(-64\)\ < \ 0\)]], "and ", Cell[BoxData[ \(TraditionalForm\`d(5, 0)\ = \(-64\) < \ 0\)]], ", the graph of ", Cell[BoxData[ \(TraditionalForm\`f\)]], " curves in different directions at each of these points and thus ", Cell[BoxData[ \(TraditionalForm\`f\)]], " has saddle points at ", Cell[BoxData[ \(TraditionalForm\`\((\(-3\), 0)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\((5, 0)\)\)]], ".\nSince ", Cell[BoxData[ \(TraditionalForm\`d(1, 4) = \ 32 > 0\)]], "and ", Cell[BoxData[ \(TraditionalForm\`\(f\_xx\)(1, 4)\ = \ 8\ > \ 0\)]], ", the graph of ", Cell[BoxData[ \(TraditionalForm\`f\)]], " is curved up in all d irections at ", Cell[BoxData[ \(TraditionalForm\`\((1, 4)\)\)]], " and therefore has a local mininum there, ", Cell[BoxData[ \(TraditionalForm\`f(1, 4)\ = \ \(-32\)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot3D[f[x, y], {x, \(-5\), 7}, {y, \(-5\), 10}, \ BoxRatios \[Rule] {1, 1, 2}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.62453 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind 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\(D[f[x, y], y]\)], "Input"], Cell[BoxData[ \(\(-6\)\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x, y], {x, 2}]\)], "Input"], Cell[BoxData[ \(\(-12\) + 6\ x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x, y], {y, 2}]\)], "Input"], Cell[BoxData[ \(\(-6\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x, y], x, y]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["The crititcal points are:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cps\ = \ Solve[{D[f[x, y], x]\ \[Equal] \ 0, \ D[f[x, y], y] \[Equal] 0}]\)], "Input"], Cell[BoxData[ \({{x \[Rule] 0, y \[Rule] 0}, {x \[Rule] 4, y \[Rule] 0}}\)], "Output"] }, Open ]], Cell["\<\ Next we apply the Second Derivative Test at the criticial points.\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(d[x_, y_]\ = \ D[f[x, y], {x, 2}] D[f[x, y], {y, 2}] - D[f[x, y], x, y]^2\)], "Input"], Cell[BoxData[ \(\(-6\)\ \((\(-12\) + 6\ x)\)\)], "Output"] }, Open ]], 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Cell[BoxData[ \(TraditionalForm\`f\)]], " curves in different directions at this point and ", Cell[BoxData[ \(TraditionalForm\`f\)]], " has a saddle point at ", Cell[BoxData[ \(TraditionalForm\`\((4, 0)\)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot3D[f[x, y], {x, \(-3\), 7}, {y, \(-5\), 5}, \ BoxRatios \[Rule] {1, 1, 2}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.62453 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% SurfaceGraphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.30443 1.63518 -0.0106411 1.63518 [ [.43557 1.6305 -6.06615 0 ] [.43557 1.6305 5.93385 9 ] [.5562 1.60104 -2.98157 0 ] [.5562 1.60104 3.01843 9 ] [.67967 1.57089 -2.92841 0 ] [.67967 1.57089 3.07159 9 ] [.80608 1.54001 -2.87351 0 ] [.80608 1.54001 3.12649 9 ] [.93554 1.50838 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m .34289 1.58746 L s [(4)] .34214 1.61811 .02434 -1 Mshowa .125 Mabswid .06207 1.37643 m .06248 1.37031 L s .08228 1.39237 m .08267 1.38625 L s .10226 1.40813 m .10263 1.40201 L s .14155 1.4391 m .14188 1.43298 L s .16086 1.45433 m .16117 1.44821 L s .17995 1.46939 m .18025 1.46326 L s .21751 1.499 m .21777 1.49288 L s .23597 1.51357 m .23622 1.50744 L s .25424 1.52797 m .25447 1.52184 L s .29017 1.55631 m .29037 1.55018 L s .30785 1.57025 m .30803 1.56412 L s .32534 1.58404 m .3255 1.57791 L s .02093 1.34399 m .02138 1.33787 L s .35975 1.61118 m .35989 1.60505 L s .25 Mabswid .0843 .2212 m 0 1.32748 L s .08062 .2694 m .09015 .2657 L s [(-150)] .06157 .27679 1 -0.38783 Mshowa .06331 .4966 m .07291 .49309 L s [(-100)] .04412 .50362 1 -0.36572 Mshowa .04456 .74274 m .05423 .73944 L s [(-50)] .02521 .74935 1 -0.34168 Mshowa .02417 1.0103 m .03391 1.00723 L s [(0)] .00468 1.01645 1 -0.31544 Mshowa .00193 1.30221 m .01175 1.29939 L s [(50)] -0.01772 1.30784 1 -0.28669 Mshowa .125 Mabswid 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1 1.47188 L s 1 1.47188 m .37669 1.62453 L s .37669 1.62453 m .3971 .57309 L s 0 0 m 1 0 L 1 1.62453 L 0 1.62453 L closepath clip newpath .5 Mabswid .828 .517 .464 r .3742 .65614 .39668 .59446 .43053 .74434 .40806 .81116 Metetra .776 .457 .449 r .40806 .81116 .43053 .74434 .46696 .84294 .44479 .91339 Metetra .694 .37 .423 r .44479 .91339 .46696 .84294 .50508 .89421 .48344 .9668 Metetra .573 .249 .382 r .48344 .9668 .50508 .89421 .54407 .90524 .52314 .97864 Metetra .419 .106 .326 r .52314 .97864 .54407 .90524 .5833 .88526 .56319 .9584 Metetra .271 0 .27 r .56319 .9584 .5833 .88526 .62236 .84458 .60314 .91674 Metetra .183 0 .234 r .60314 .91674 .62236 .84458 .66112 .79377 .64282 .86463 Metetra .183 0 .234 r .64282 .86463 .66112 .79377 .6997 .74321 .68229 .81277 Metetra .269 0 .268 r .68229 .81277 .6997 .74321 .73846 .70286 .72193 .77147 Metetra .412 .097 .318 r .72193 .77147 .73846 .70286 .77802 .68245 .76233 .75077 Metetra .564 .232 .365 r .76233 .75077 .77802 .68245 .81929 .69197 .80441 .76103 Metetra .689 .35 .396 r .80441 .76103 .81929 .69197 .86351 .74253 .84942 .81375 Metetra .778 .439 .413 r .84942 .81375 .86351 .74253 .9124 .84764 .89912 .923 Metetra .837 .501 .418 r .89912 .923 .9124 .84764 .9684 1.02538 .956 1.1077 Metetra .845 .528 .453 r .35112 .70683 .3742 .65614 .40806 .81116 .3849 .8665 Metetra .795 .467 .433 r .3849 .8665 .40806 .81116 .44479 .91339 .42185 .97202 Metetra .711 .371 .4 r .42185 .97202 .44479 .91339 .48344 .9668 .46095 1.02736 Metetra .573 .227 .345 r .46095 1.02736 .48344 .9668 .52314 .97864 .50128 1.0399 Metetra .378 .042 .265 r .50128 1.0399 .52314 .97864 .56319 .9584 .54209 1.01941 Metetra .178 0 .179 r .54209 1.01941 .56319 .9584 .60314 .91674 .58287 .97686 Metetra .057 0 .126 r .58287 .97686 .60314 .91674 .64282 .86463 .62341 .92353 Metetra .057 0 .126 r .62341 .92353 .64282 .86463 .68229 .81277 .66374 .87047 Metetra .175 0 .176 r .66374 .87047 .68229 .81277 .72193 .77147 .70419 .82828 Metetra .368 .029 .252 r .70419 .82828 .72193 .77147 .76233 .75077 .74537 .80732 Metetra .562 .206 .322 r .74537 .80732 .76233 .75077 .80441 .76103 .78818 .81823 Metetra .706 .348 .367 r .78818 .81823 .80441 .76103 .84942 .81375 .83388 .87289 Metetra .799 .446 .39 r .83388 .87289 .84942 .81375 .89912 .923 .88427 .98591 Metetra .855 .511 .399 r .88427 .98591 .89912 .923 .956 1.1077 .94186 1.17697 Metetra .863 .541 .439 r .32757 .74556 .35112 .70683 .3849 .8665 .36118 .90928 Metetra .818 .478 .413 r .36118 .90928 .3849 .8665 .42185 .97202 .39824 1.01768 Metetra .733 .373 .369 r .39824 1.01768 .42185 .97202 .46095 1.02736 .4377 1.07469 Metetra .572 .195 .29 r .4377 1.07469 .46095 1.02736 .50128 1.0399 .47859 1.08784 Metetra .307 0 .162 r .47859 1.08784 .50128 1.0399 .54209 1.01941 .52009 1.0671 Metetra .018 0 .024 r .52009 1.0671 .54209 1.01941 .58287 .97686 .56164 1.02373 Metetra .56164 1.02373 .58287 .97686 .62341 .92353 .60297 .96931 Metetra .60297 .96931 .62341 .92353 .66374 .87047 .64409 .91516 Metetra .012 0 .019 r .64409 .91516 .66374 .87047 .70419 .82828 .6853 .87217 Metetra .291 0 .143 r .6853 .87217 .70419 .82828 .74537 .80732 .72718 .85094 Metetra .556 .165 .256 r .72718 .85094 .74537 .80732 .78818 .81823 .77063 .86239 Metetra .727 .344 .325 r .77063 .86239 .78818 .81823 .83388 .87289 .81693 .91875 Metetra .822 .455 .36 r .81693 .91875 .83388 .87289 .88427 .98591 .86786 1.03507 Metetra .875 .521 .376 r .86786 1.03507 .88427 .98591 .94186 1.17697 .926 1.23175 Metetra .883 .554 .422 r .3037 .77144 .32757 .74556 .36118 .90928 .33704 .93854 Metetra .843 .491 .387 r .33704 .93854 .36118 .90928 .39824 1.01768 .37411 1.04933 Metetra .759 .374 .324 r .37411 1.04933 .39824 1.01768 .4377 1.07469 .41384 1.10773 Metetra .566 .143 .198 r .41384 1.10773 .4377 1.07469 .47859 1.08784 .4552 1.12136 Metetra .17 0 0 r .4552 1.12136 .47859 1.08784 .52009 1.0671 .49731 1.10037 Metetra .49731 1.10037 .52009 1.0671 .56164 1.02373 .53955 1.05627 Metetra .53955 1.05627 .56164 1.02373 .60297 .96931 .5816 1.00088 Metetra .5816 1.00088 .60297 .96931 .64409 .91516 .62344 .94579 Metetra .62344 .94579 .64409 .91516 .6853 .87217 .66533 .90207 Metetra .141 0 0 r .66533 .90207 .6853 .87217 .72718 .85094 .70783 .88058 Metetra .541 .097 .144 r .70783 .88058 .72718 .85094 .77063 .86239 .75183 .89247 Metetra .752 .337 .263 r .75183 .89247 .77063 .86239 .81693 .91875 .7986 .95022 Metetra .848 .463 .32 r .7986 .95022 .81693 .91875 .86786 1.03507 .84993 1.06929 Metetra .895 .532 .346 r .84993 1.06929 .86786 1.03507 .926 1.23175 .90842 1.27071 Metetra .903 .568 .4 r .27964 .7837 .3037 .77144 .33704 .93854 .31261 .95342 Metetra .87 .504 .352 r .31261 .95342 .33704 .93854 .37411 1.04933 .34961 1.06606 Metetra .79 .372 .256 r .34961 1.06606 .37411 1.04933 .41384 1.10773 .38952 1.12552 Metetra .54 .043 .024 r .38952 1.12552 .41384 1.10773 .4552 1.12136 .43125 1.13948 Metetra .43125 1.13948 .4552 1.12136 .49731 1.10037 .47388 1.11826 Metetra .669 .932 .644 r .47388 1.11826 .49731 1.10037 .53955 1.05627 .51673 1.07352 Metetra .812 .972 .668 r .51673 1.07352 .53955 1.05627 .5816 1.00088 .55942 1.01733 Metetra .812 .972 .668 r .55942 1.01733 .5816 1.00088 .62344 .94579 .60189 .96143 Metetra .676 .936 .651 r .60189 .96143 .62344 .94579 .66533 .90207 .64437 .9171 Metetra .64437 .9171 .66533 .90207 .70783 .88058 .68741 .89535 Metetra .492 0 0 r .68741 .89535 .70783 .88058 .75183 .89247 .73186 .90752 Metetra .777 .322 .165 r .73186 .90752 .75183 .89247 .7986 .95022 .77898 .96633 Metetra .873 .47 .264 r .77898 .96633 .7986 .95022 .84993 1.06929 .83055 1.08754 Metetra .913 .542 .309 r .83055 1.08754 .84993 1.06929 .90842 1.27071 .88918 1.29269 Metetra .923 .582 .373 r .25556 .78173 .27964 .7837 .31261 .95342 .28808 .95324 Metetra .897 .517 .303 r .28808 .95324 .31261 .95342 .34961 1.06606 .32491 1.06715 Metetra .817 .36 .144 r .32491 1.06715 .34961 1.06606 .38952 1.12552 .36489 1.1273 Metetra .424 0 0 r .36489 1.1273 .38952 1.12552 .43125 1.13948 .4069 1.14143 Metetra .584 .944 .9 r .4069 1.14143 .43125 1.13948 .47388 1.11826 .44995 1.11997 Metetra .871 .981 .842 r .44995 1.11997 .47388 1.11826 .51673 1.07352 .49331 1.07472 Metetra .92 .956 .795 r .49331 1.07472 .51673 1.07352 .55942 1.01733 .53655 1.01788 Metetra .92 .956 .795 r .53655 1.01788 .55942 1.01733 .60189 .96143 .57957 .96134 Metetra .871 .98 .843 r .57957 .96134 .60189 .96143 .64437 .9171 .62256 .9165 Metetra .615 .952 .919 r .62256 .9165 .64437 .9171 .68741 .89535 .66602 .8945 Metetra .305 0 0 r .66602 .8945 .68741 .89535 .73186 .90752 .71081 .90681 Metetra .782 .283 .003 r .71081 .90681 .73186 .90752 .77898 .96633 .75815 .96629 Metetra .891 .472 .187 r .75815 .96629 .77898 .96633 .83055 1.08754 .80981 1.08896 Metetra .927 .549 .262 r .80981 1.08896 .83055 1.08754 .88918 1.29269 .86837 1.29673 Metetra .94 .595 .339 r .23164 .7651 .25556 .78173 .28808 .95324 .26362 .93751 Metetra .917 .526 .235 r .26362 .93751 .28808 .95324 .32491 1.06715 .30018 1.05204 Metetra .806 .319 0 r .30018 1.05204 .32491 1.06715 .36489 1.1273 .34013 1.11249 Metetra 0 .463 .833 r .34013 1.11249 .36489 1.1273 .4069 1.14143 .38231 1.12663 Metetra .725 .863 .946 r .38231 1.12663 .4069 1.14143 .44995 1.11997 .42568 1.10493 Metetra .861 .858 .836 r .42568 1.10493 .44995 1.11997 .49331 1.07472 .46946 1.05929 Metetra .892 .846 .792 r .46946 1.05929 .49331 1.07472 .53655 1.01788 .51316 1.00198 Metetra .892 .846 .792 r .51316 1.00198 .53655 1.01788 .57957 .96134 .55662 .94497 Metetra .861 .856 .835 r .55662 .94497 .57957 .96134 .62256 .9165 .60002 .89975 Metetra .729 .853 .94 r .60002 .89975 .62256 .9165 .66602 .8945 .64381 .8775 Metetra .1 .546 .912 r .64381 .8775 .66602 .8945 .71081 .90681 .68881 .88978 Metetra .702 .193 0 r .68881 .88978 .71081 .90681 .75815 .96629 .73624 .94954 Metetra .886 .46 .079 r .73624 .94954 .75815 .96629 .80981 1.08896 .78782 1.07293 Metetra .932 .551 .203 r .78782 1.07293 .80981 1.08896 .86837 1.29673 .84609 1.28212 Metetra .953 .605 .295 r .20804 .73355 .23164 .7651 .26362 .93751 .23941 .90595 Metetra .917 .523 .14 r .23941 .90595 .26362 .93751 .30018 1.05204 .27561 1.02043 Metetra .672 .213 0 r .27561 1.02043 .30018 1.05204 .34013 1.11249 .31544 1.08078 Metetra .276 .521 .907 r .31544 1.08078 .34013 1.11249 .38231 1.12663 .35769 1.09474 Metetra .71 .722 .859 r .35769 1.09474 .38231 1.12663 .42568 1.10493 .40127 1.07281 Metetra .812 .745 .789 r .40127 1.07281 .42568 1.10493 .46946 1.05929 .44535 1.0269 Metetra .842 .749 .76 r .44535 1.0269 .46946 1.05929 .51316 1.00198 .4894 .9693 Metetra .842 .749 .76 r .4894 .9693 .51316 1.00198 .55662 .94497 .53321 .91201 Metetra .812 .744 .789 r .53321 .91201 .55662 .94497 .60002 .89975 .5769 .86653 Metetra .711 .717 .854 r .5769 .86653 .60002 .89975 .64381 .8775 .62091 .84405 Metetra .347 .544 .904 r .62091 .84405 .64381 .8775 .68881 .88978 .66602 .85612 Metetra 0 0 .513 r .66602 .85612 .68881 .88978 .73624 .94954 .71339 .91574 Metetra .834 .421 0 r .71339 .91574 .73624 .94954 .78782 1.07293 .76472 1.03908 Metetra .921 .544 .13 r .76472 1.03908 .78782 1.07293 .84609 1.28212 .82251 1.24842 Metetra .954 .61 .24 r .18494 .68703 .20804 .73355 .23941 .90595 .21564 .85847 Metetra .873 .495 .014 r .21564 .85847 .23941 .90595 .27561 1.02043 .25139 .97223 Metetra 0 0 .525 r .25139 .97223 .27561 1.02043 .31544 1.08078 .291 1.03206 Metetra .401 .482 .828 r .291 1.03206 .31544 1.08078 .35769 1.09474 .33321 1.04568 Metetra .682 .629 .789 r .33321 1.04568 .35769 1.09474 .40127 1.07281 .37689 1.02352 Metetra .769 .665 .748 r .37689 1.02352 .40127 1.07281 .44535 1.0269 .42116 .97747 Metetra .798 .676 .729 r .42116 .97747 .44535 1.0269 .4894 .9693 .46545 .91978 Metetra .798 .676 .729 r .46545 .91978 .4894 .9693 .53321 .91201 .5095 .8624 Metetra .769 .665 .748 r .5095 .8624 .53321 .91201 .5769 .86653 .55338 .81678 Metetra .683 .627 .787 r .55338 .81678 .5769 .86653 .62091 .84405 .5975 .79407 Metetra .436 .496 .823 r .5975 .79407 .62091 .84405 .66602 .85612 .64259 .80577 Metetra 0 .092 .649 r .64259 .80577 .66602 .85612 .71339 .91574 .68977 .86481 Metetra .715 .347 0 r .68977 .86481 .71339 .91574 .76472 1.03908 .74069 .98733 Metetra .885 .524 .043 r .74069 .98733 .76472 1.03908 .82251 1.24842 .79777 1.19552 Metetra .94 .605 .173 r .16252 .62573 .18494 .68703 .21564 .85847 .19248 .79527 Metetra .766 .431 0 r .19248 .79527 .21564 .85847 .25139 .97223 .22772 .90763 Metetra 0 .067 .619 r .22772 .90763 .25139 .97223 .291 1.03206 .267 .96653 Metetra .451 .448 .764 r .267 .96653 .291 1.03206 .33321 1.04568 .30906 .97961 Metetra .659 .568 .741 r .30906 .97961 .33321 1.04568 .37689 1.02352 .35274 .95724 Metetra .736 .608 .716 r .35274 .95724 .37689 1.02352 .42116 .97747 .39709 .91118 Metetra .764 .622 .703 r .39709 .91118 .42116 .97747 .46545 .91978 .4415 .8536 Metetra .764 .622 .703 r .4415 .8536 .46545 .91978 .5095 .8624 .48567 .79632 Metetra .736 .608 .716 r .48567 .79632 .5095 .8624 .55338 .81678 .52963 .7507 Metetra .66 .568 .741 r .52963 .7507 .55338 .81678 .5975 .79407 .57374 .72777 Metetra .472 .46 .763 r .57374 .72777 .5975 .79407 .64259 .80577 .61869 .73892 Metetra .03 .173 .679 r .61869 .73892 .64259 .80577 .68977 .86481 .66555 .79697 Metetra 0 0 .352 r .66555 .79697 .68977 .86481 .74069 .98733 .71591 .91786 Metetra .82 .486 0 r .71591 .91786 .74069 .98733 .79777 1.19552 .77209 1.12362 Metetra .902 .586 .094 r .14093 .55002 .16252 .62573 .19248 .79527 .17011 .71675 Metetra 0 0 .263 r .17011 .71675 .19248 .79527 .22772 .90763 .20476 .82705 Metetra .051 .141 .639 r .20476 .82705 .22772 .90763 .267 .96653 .24364 .88462 Metetra .475 .423 .72 r .24364 .88462 .267 .96653 .30906 .97961 .28545 .89699 Metetra .641 .526 .707 r .28545 .89699 .30906 .97961 .35274 .95724 .329 .87441 Metetra .71 .566 .691 r .329 .87441 .35274 .95724 .39709 .91118 .37332 .82849 Metetra .736 .582 .682 r .37332 .82849 .39709 .91118 .4415 .8536 .41774 .7712 Metetra .736 .582 .682 r .41774 .7712 .4415 .8536 .48567 .79632 .46192 .71422 Metetra .71 .566 .691 r .46192 .71422 .48567 .79632 .52963 .7507 .50584 .66873 Metetra .643 .527 .707 r .50584 .66873 .52963 .7507 .57374 .72777 .54983 .64559 Metetra .49 .434 .721 r .54983 .64559 .57374 .72777 .61869 .73892 .59452 .65604 Metetra .157 .216 .674 r .59452 .65604 .61869 .73892 .66555 .79697 .64093 .71267 Metetra 0 0 .447 r .64093 .71267 .66555 .79697 .71591 .91786 .69057 .83119 Metetra .727 .431 0 r .69057 .83119 .71591 .91786 .77209 1.12362 .74568 1.03324 Metetra .837 .551 .008 r .12033 .4605 .14093 .55002 .17011 .71675 .1487 .62355 Metetra 0 0 .365 r .1487 .62355 .17011 .71675 .20476 .82705 .18269 .73117 Metetra .162 .184 .636 r .18269 .73117 .20476 .82705 .24364 .88462 .22108 .78701 Metetra .488 .405 .688 r .22108 .78701 .24364 .88462 .28545 .89699 .26254 .79853 Metetra .628 .495 .682 r .26254 .79853 .28545 .89699 .329 .87441 .30586 .77575 Metetra .689 .534 .671 r .30586 .77575 .329 .87441 .37332 .82849 .35003 .73009 Metetra .714 .55 .665 r .35003 .73009 .37332 .82849 .41774 .7712 .39434 .67329 Metetra .714 .55 .665 r .39434 .67329 .41774 .7712 .46192 .71422 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y], y]\)], "Input"], Cell[BoxData[ \(\(-2\)\ x + 2\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x, y], {x, 2}]\)], "Input"], Cell[BoxData[ \(2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x, y], {y, 2}]\)], "Input"], Cell[BoxData[ \(2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x, y], x, y]\)], "Input"], Cell[BoxData[ \(\(-2\)\)], "Output"] }, Open ]], Cell["The crititcal points are:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cps\ = \ Solve[{D[f[x, y], x]\ \[Equal] \ 0, \ D[f[x, y], y] \[Equal] 0}]\)], "Input"], Cell[BoxData[ \(Solve::"svars" \(\(:\)\(\ \)\) "Equations may not give solutions for all \"solve\" variables."\)], \ "Message"], Cell[BoxData[ \({{x \[Rule] y}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(d[x_, y_]\ = \ D[f[x, y], {x, 2}] D[f[x, y], {y, 2}] - D[f[x, y], x, y]^2\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f[x, y]\ /. \ 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1.56264 .03466 -1 Mshowa .34264 1.59768 m .34289 1.58746 L s [(4)] .34214 1.61811 .02434 -1 Mshowa .125 Mabswid .06207 1.37643 m .06248 1.37031 L s .08228 1.39237 m .08267 1.38625 L s .10226 1.40813 m .10263 1.40201 L s .14155 1.4391 m .14188 1.43298 L s .16086 1.45433 m .16117 1.44821 L s .17995 1.46939 m .18025 1.46326 L s .21751 1.499 m .21777 1.49288 L s .23597 1.51357 m .23622 1.50744 L s .25424 1.52797 m .25447 1.52184 L s .29017 1.55631 m .29037 1.55018 L s .30785 1.57025 m .30803 1.56412 L s .32534 1.58404 m .3255 1.57791 L s .02093 1.34399 m .02138 1.33787 L s .35975 1.61118 m .35989 1.60505 L s .25 Mabswid .0843 .2212 m 0 1.32748 L s .08261 .2433 m .09213 .23959 L s [(0)] .06357 .25073 1 -0.39037 Mshowa .06502 .47423 m .07461 .4707 L s [(25)] .04583 .48129 1 -0.3679 Mshowa .04592 .72484 m .05559 .72152 L s [(50)] .02659 .73148 1 -0.34343 Mshowa .02512 .99776 m .03487 .99468 L s [(75)] .00564 1.00394 1 -0.31667 Mshowa .00239 1.29611 m .01221 1.29329 L s [(100)] -0.01725 1.30176 1 -0.2873 Mshowa .125 Mabswid .07921 .28803 m .08493 .28582 L s .07574 .33346 m .08147 .33128 L s .07222 .37963 m .07796 .37747 L s .06865 .42655 m .0744 .42441 L s .06132 .5227 m .06709 .52061 L s .05757 .57198 m .06334 .56991 L s .05375 .62208 m .05953 .62004 L s .04987 .67303 m .05566 .67101 L s .0419 .77755 m .04771 .77559 L s .03782 .83117 m .04364 .82924 L s .03366 .88573 m .03949 .88382 L s .02943 .94125 m .03527 .93937 L s .02074 1.05529 m .0266 1.05347 L s .01628 1.11386 m .02214 1.11207 L s .01173 1.1735 m .01761 1.17174 L s .0071 1.23424 m .01299 1.23251 L s .25 Mabswid .0843 .2212 m 0 1.32748 L s 0 1.32748 m .37669 1.62453 L s .37669 1.62453 m .3971 .57309 L s .3971 .57309 m .0843 .2212 L s .64785 0 m .92348 .39072 L s .92348 .39072 m 1 1.47188 L s 1 1.47188 m .67613 1.13533 L s .67613 1.13533 m .64785 0 L s .0843 .2212 m 0 1.32748 L s 0 1.32748 m .67613 1.13533 L s .67613 1.13533 m .64785 0 L s .64785 0 m .0843 .2212 L s .3971 .57309 m .92348 .39072 L s .92348 .39072 m 1 1.47188 L s 1 1.47188 m .37669 1.62453 L s .37669 1.62453 m .3971 .57309 L s 0 0 m 1 0 L 1 1.62453 L 0 1.62453 L closepath clip newpath .5 Mabswid .752 .543 .613 r .35674 1.41799 .37726 1.59522 .42135 1.42719 .40036 1.26669 Metetra .752 .545 .617 r .40036 1.26669 .42135 1.42719 .46376 1.27624 .44244 1.13095 Metetra .751 .548 .621 r .44244 1.13095 .46376 1.27624 .50477 1.1408 .48324 1.00943 Metetra .75 .551 .627 r .48324 1.00943 .50477 1.1408 .54463 1.01955 .52302 .901 Metetra .749 .555 .633 r .52302 .901 .54463 1.01955 .58359 .91135 .56198 .80472 Metetra .747 .56 .641 r .56198 .80472 .58359 .91135 .62185 .81528 .60034 .71979 Metetra .745 .565 .651 r .60034 .71979 .62185 .81528 .65961 .73053 .63829 .64555 Metetra .742 .572 .662 r .63829 .64555 .65961 .73053 .69707 .65645 .676 .58145 Metetra .739 .58 .677 r .676 .58145 .69707 .65645 .73439 .5925 .71367 .52705 Metetra .733 .59 .696 r .71367 .52705 .73439 .5925 .77175 .53823 .75145 .48201 Metetra .725 .604 .722 r .75145 .48201 .77175 .53823 .80932 .49331 .78954 .44607 Metetra .71 .621 .756 r .78954 .44607 .80932 .49331 .84727 .45748 .82808 .41906 Metetra .685 .645 .804 r .82808 .41906 .84727 .45748 .88578 .43057 .86728 .4009 Metetra .631 .675 .873 r .86728 .4009 .88578 .43057 .92502 .4125 .90732 .39158 Metetra .752 .545 .616 r .33654 1.25708 .35674 1.41799 .40036 1.26669 .3797 1.12103 Metetra .751 .548 .621 r .3797 1.12103 .40036 1.26669 .44244 1.13095 .42146 .99925 Metetra .75 .551 .627 r .42146 .99925 .44244 1.13095 .48324 1.00943 .46206 .89059 Metetra .749 .555 .633 r .46206 .89059 .48324 1.00943 .52302 .901 .50173 .7941 Metetra .747 .56 .641 r .50173 .7941 .52302 .901 .56198 .80472 .54069 .70899 Metetra .745 .565 .651 r .54069 .70899 .56198 .80472 .60034 .71979 .57914 .63458 Metetra .742 .572 .662 r .57914 .63458 .60034 .71979 .63829 .64555 .61726 .57033 Metetra .739 .58 .677 r .61726 .57033 .63829 .64555 .676 .58145 .65523 .5158 Metetra .733 .59 .696 r .65523 .5158 .676 .58145 .71367 .52705 .69323 .47064 Metetra .725 .604 .722 r .69323 .47064 .71367 .52705 .75145 .48201 .73144 .43459 Metetra .711 .622 .756 r .73144 .43459 .75145 .48201 .78954 .44607 .77003 .40749 Metetra .685 .645 .805 r .77003 .40749 .78954 .44607 .82808 .41906 .80918 .38923 Metetra .631 .675 .874 r .80918 .38923 .82808 .41906 .86728 .4009 .84908 .37983 Metetra .496 .696 .958 r .84908 .37983 .86728 .4009 .90732 .39158 .88993 .37935 Metetra .751 .548 .621 r .31655 1.11105 .33654 1.25708 .3797 1.12103 .35927 .989 Metetra .75 .551 .627 r .35927 .989 .3797 1.12103 .42146 .99925 .4007 .8801 Metetra .749 .555 .633 r .4007 .8801 .42146 .99925 .46206 .89059 .44109 .78341 Metetra .747 .56 .641 r .44109 .78341 .46206 .89059 .50173 .7941 .48066 .69811 Metetra .745 .565 .651 r .48066 .69811 .50173 .7941 .54069 .70899 .51962 .62354 Metetra .743 .572 .662 r .51962 .62354 .54069 .70899 .57914 .63458 .55814 .55915 Metetra .739 .58 .677 r .55814 .55915 .57914 .63458 .61726 .57033 .59642 .50448 Metetra .733 .59 .696 r .59642 .50448 .61726 .57033 .65523 .5158 .63465 .4592 Metetra .725 .604 .722 r .63465 .4592 .65523 .5158 .69323 .47064 .67298 .42304 Metetra .711 .622 .756 r .67298 .42304 .69323 .47064 .73144 .43459 .71161 .39583 Metetra .685 .646 .805 r .71161 .39583 .73144 .43459 .77003 .40749 .75072 .37749 Metetra .631 .676 .874 r .75072 .37749 .77003 .40749 .80918 .38923 .79048 .368 Metetra .495 .696 .958 r .79048 .368 .80918 .38923 .84908 .37983 .8311 .36745 Metetra .135 .594 .925 r .8311 .36745 .84908 .37983 .88993 .37935 .87278 .37598 Metetra .75 .551 .627 r .29667 .97868 .31655 1.11105 .35927 .989 .33895 .86955 Metetra .749 .555 .633 r .33895 .86955 .35927 .989 .4007 .8801 .38007 .77265 Metetra .747 .559 .641 r .38007 .77265 .4007 .8801 .44109 .78341 .42026 .68717 Metetra .745 .565 .65 r .42026 .68717 .44109 .78341 .48066 .69811 .45972 .61244 Metetra .743 .572 .662 r .45972 .61244 .48066 .69811 .51962 .62354 .49865 .54789 Metetra .739 .58 .677 r .49865 .54789 .51962 .62354 .55814 .55915 .53725 .49309 Metetra .733 .59 .696 r .53725 .49309 .55814 .55915 .59642 .50448 .5757 .44769 Metetra .725 .604 .722 r .5757 .44769 .59642 .50448 .63465 .4592 .61416 .41142 Metetra .711 .622 .756 r .61416 .41142 .63465 .4592 .67298 .42304 .65283 .38411 Metetra .685 .646 .805 r .65283 .38411 .67298 .42304 .71161 .39583 .69189 .36567 Metetra .63 .676 .875 r .69189 .36567 .71161 .39583 .75072 .37749 .73151 .3561 Metetra .493 .696 .959 r .73151 .3561 .75072 .37749 .79048 .368 .7719 .35547 Metetra .131 .592 .923 r .7719 .35547 .79048 .368 .8311 .36745 .81325 .36394 Metetra .375 0 0 r .81325 .36394 .8311 .36745 .87278 .37598 .8558 .38177 Metetra .749 .555 .633 r .2768 .85893 .29667 .97868 .33895 .86955 .31865 .76183 Metetra .747 .559 .641 r .31865 .76183 .33895 .86955 .38007 .77265 .35946 .67616 Metetra .745 .565 .65 r .35946 .67616 .38007 .77265 .42026 .68717 .39944 .60126 Metetra .743 .572 .662 r .39944 .60126 .42026 .68717 .45972 .61244 .43879 .53657 Metetra .739 .58 .677 r .43879 .53657 .45972 .61244 .49865 .54789 .47771 .48163 Metetra .733 .59 .696 r .47771 .48163 .49865 .54789 .53725 .49309 .51638 .4361 Metetra .725 .604 .722 r .51638 .4361 .53725 .49309 .5757 .44769 .55497 .39972 Metetra .711 .622 .756 r .55497 .39972 .5757 .44769 .61416 .41142 .59368 .37231 Metetra .685 .646 .805 r .59368 .37231 .61416 .41142 .65283 .38411 .63269 .35378 Metetra .63 .676 .875 r .63269 .35378 .65283 .38411 .69189 .36567 .67217 .34413 Metetra .492 .696 .959 r .67217 .34413 .69189 .36567 .73151 .3561 .71232 .34342 Metetra .127 .59 .921 r .71232 .34342 .73151 .3561 .7719 .35547 .75335 .35183 Metetra .379 0 0 r .75335 .35183 .7719 .35547 .81325 .36394 .79547 .3696 Metetra .619 .069 0 r .79547 .3696 .81325 .36394 .8558 .38177 .83892 .39709 Metetra .747 .559 .641 r .25684 .75093 .2768 .85893 .31865 .76183 .29828 .66508 Metetra .745 .565 .65 r .29828 .66508 .31865 .76183 .35946 .67616 .33878 .59001 Metetra .743 .572 .662 r .33878 .59001 .35946 .67616 .39944 .60126 .37855 .52517 Metetra .739 .58 .677 r .37855 .52517 .39944 .60126 .43879 .53657 .41779 .4701 Metetra .733 .59 .696 r .41779 .4701 .43879 .53657 .47771 .48163 .45668 .42445 Metetra .725 .604 .722 r .45668 .42445 .47771 .48163 .51638 .4361 .49541 .38795 Metetra .711 .622 .756 r .49541 .38795 .51638 .4361 .55497 .39972 .53416 .36044 Metetra .685 .646 .806 r .53416 .36044 .55497 .39972 .59368 .37231 .57311 .34182 Metetra .63 .676 .876 r .57311 .34182 .59368 .37231 .63269 .35378 .61245 .33208 Metetra .491 .696 .96 r .61245 .33208 .63269 .35378 .67217 .34413 .65236 .3313 Metetra .122 .588 .919 r .65236 .3313 .67217 .34413 .71232 .34342 .69305 .33963 Metetra .383 0 0 r .69305 .33963 .71232 .34342 .75335 .35183 .73474 .35735 Metetra .621 .072 0 r .73474 .35735 .75335 .35183 .79547 .3696 .77765 .38479 Metetra .7 .215 .091 r .77765 .38479 .79547 .3696 .83892 .39709 .82204 .42243 Metetra .745 .565 .65 r .23671 .65392 .25684 .75093 .29828 .66508 .27774 .57869 Metetra .743 .572 .662 r .27774 .57869 .29828 .66508 .33878 .59001 .31793 .5137 Metetra .739 .58 .677 r .31793 .5137 .33878 .59001 .37855 .52517 .35749 .45849 Metetra .733 .59 .696 r .35749 .45849 .37855 .52517 .41779 .4701 .39661 .41272 Metetra .725 .604 .722 r .39661 .41272 .41779 .4701 .45668 .42445 .43547 .37611 Metetra .711 .622 .757 r .43547 .37611 .45668 .42445 .49541 .38795 .47426 .34849 Metetra .684 .646 .806 r .47426 .34849 .49541 .38795 .53416 .36044 .51316 .32978 Metetra .629 .677 .876 r .51316 .32978 .53416 .36044 .57311 .34182 .55235 .31995 Metetra .489 .696 .96 r .55235 .31995 .57311 .34182 .61245 .33208 .59202 .31909 Metetra .118 .585 .917 r .59202 .31909 .61245 .33208 .65236 .3313 .63237 .32736 Metetra .386 0 0 r .63237 .32736 .65236 .3313 .69305 .33963 .67362 .34502 Metetra .622 .074 0 r .67362 .34502 .69305 .33963 .73474 .35735 .71598 .37241 Metetra .701 .216 .093 r .71598 .37241 .73474 .35735 .77765 .38479 .75972 .41001 Metetra .731 .29 .204 r .75972 .41001 .77765 .38479 .82204 .42243 .8051 .45842 Metetra .743 .572 .662 r .2163 .5673 .23671 .65392 .27774 .57869 .25692 .50215 Metetra .739 .58 .677 r .25692 .50215 .27774 .57869 .31793 .5137 .29681 .44681 Metetra .733 .59 .696 r .29681 .44681 .31793 .5137 .35749 .45849 .33616 .40091 Metetra .725 .604 .722 r .33616 .40091 .35749 .45849 .39661 .41272 .37516 .36419 Metetra .711 .622 .757 r .37516 .36419 .39661 .41272 .43547 .37611 .41398 .33647 Metetra .684 .646 .806 r .41398 .33647 .43547 .37611 .47426 .34849 .45282 .31766 Metetra .629 .677 .877 r .45282 .31766 .47426 .34849 .51316 .32978 .49186 .30775 Metetra .488 .696 .961 r .49186 .30775 .51316 .32978 .55235 .31995 .53129 .30681 Metetra .113 .583 .915 r .53129 .30681 .55235 .31995 .59202 .31909 .5713 .31501 Metetra .39 0 0 r .5713 .31501 .59202 .31909 .63237 .32736 .6121 .3326 Metetra .624 .076 0 r .6121 .3326 .63237 .32736 .67362 .34502 .65391 .35995 Metetra .701 .217 .095 r .65391 .35995 .67362 .34502 .71598 .37241 .69698 .39751 Metetra .732 .291 .205 r .69698 .39751 .71598 .37241 .75972 .41001 .74157 .4459 Metetra .746 .335 .271 r .74157 .4459 .75972 .41001 .8051 .45842 .78799 .50584 Metetra .739 .58 .677 r .19553 .49054 .2163 .5673 .25692 .50215 .23575 .43505 Metetra .733 .59 .696 r .23575 .43505 .25692 .50215 .29681 .44681 .27533 .38903 Metetra .725 .604 .722 r .27533 .38903 .29681 .44681 .33616 .40091 .31446 .35219 Metetra .711 .622 .757 r .31446 .35219 .33616 .40091 .37516 .36419 .35332 .32437 Metetra .684 .647 .806 r .35332 .32437 .37516 .36419 .41398 .33647 .39211 .30546 Metetra .628 .677 .877 r .39211 .30546 .41398 .33647 .45282 .31766 .43099 .29546 Metetra .486 .696 .961 r .43099 .29546 .45282 .31766 .49186 .30775 .47017 .29445 Metetra .109 .581 .913 r .47017 .29445 .49186 .30775 .53129 .30681 .50982 .30258 Metetra .394 0 0 r .50982 .30258 .53129 .30681 .5713 .31501 .55017 .32011 Metetra .625 .079 0 r .55017 .32011 .5713 .31501 .6121 .3326 .59142 .3474 Metetra .702 .219 .097 r .59142 .3474 .6121 .3326 .65391 .35995 .63382 .38493 Metetra .732 .292 .206 r .63382 .38493 .65391 .35995 .69698 .39751 .67762 .43329 Metetra .746 .335 .272 r .67762 .43329 .69698 .39751 .74157 .4459 .72311 .49323 Metetra .753 .363 .315 r .72311 .49323 .74157 .4459 .78799 .50584 .77062 .56566 Metetra .733 .59 .696 r .17429 .42323 .19553 .49054 .23575 .43505 .2141 .37707 Metetra .725 .604 .722 r .2141 .37707 .23575 .43505 .27533 .38903 .25337 .34012 Metetra .711 .622 .757 r .25337 .34012 .27533 .38903 .31446 .35219 .29227 .31219 Metetra .684 .647 .807 r .29227 .31219 .31446 .35219 .35332 .32437 .331 .29319 Metetra .628 .677 .878 r .331 .29319 .35332 .32437 .39211 .30546 .36973 .2831 Metetra .485 .696 .962 r .36973 .2831 .39211 .30546 .43099 .29546 .40865 .28201 Metetra .104 .579 .911 r .40865 .28201 .43099 .29546 .47017 .29445 .44795 .29006 Metetra .398 0 0 r .44795 .29006 .47017 .29445 .50982 .30258 .48784 .30753 Metetra .626 .081 0 r .48784 .30753 .50982 .30258 .55017 .32011 .52852 .33478 Metetra .703 .22 .099 r .52852 .33478 .55017 .32011 .59142 .3474 .57023 .37227 Metetra .732 .293 .208 r .57023 .37227 .59142 .3474 .63382 .38493 .61323 .4206 Metetra .746 .336 .273 r .61323 .4206 .63382 .38493 .67762 .43329 .65778 .48053 Metetra .753 .364 .316 r .65778 .48053 .67762 .43329 .72311 .49323 .70421 .55297 Metetra .757 .383 .346 r .70421 .55297 .72311 .49323 .77062 .56566 .75288 .63905 Metetra .725 .604 .722 r .15249 .36504 .17429 .42323 .2141 .37707 .19189 .32798 Metetra .711 .622 .757 r .19189 .32798 .2141 .37707 .25337 .34012 .23083 .29994 Metetra .684 .647 .807 r .23083 .29994 .25337 .34012 .29227 .31219 .2695 .28084 Metetra .628 .678 .878 r .2695 .28084 .29227 .31219 .331 .29319 .30807 .27066 Metetra .483 .696 .962 r .30807 .27066 .331 .29319 .36973 .2831 .34673 .26948 Metetra .1 .577 .908 r .34673 .26948 .36973 .2831 .40865 .28201 .38568 .27747 Metetra .402 0 0 r .38568 .27747 .40865 .28201 .44795 .29006 .42509 .29487 Metetra .628 .083 0 r .42509 .29487 .44795 .29006 .48784 .30753 .4652 .32206 Metetra .703 .221 .101 r .4652 .32206 .48784 .30753 .52852 .33478 .50622 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\(\(-2\) + 2\ x + 4\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f[x, y], y]\)], "Input"], Cell[BoxData[ \(4\ x + 2\ y\)], "Output"] }, Open ]], Cell["The crititcal points are:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cps\ = \ Solve[{D[f[x, y], x]\ \[Equal] \ 0, \ D[f[x, y], y] \[Equal] 0}]\)], "Input"], Cell[BoxData[ \({{x \[Rule] \(-\(1\/3\)\), y \[Rule] 2\/3}}\)], "Output"] }, Open ]], Cell["\<\ This points lies inside the region, so we check the function value \ there:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f[x, y]\ /. \ cps\)], "Input"], Cell[BoxData[ \({1\/3}\)], "Output"] }, Open ]], Cell[TextData[{ "Now we need to find the extreme values of ", Cell[BoxData[ \(TraditionalForm\`f\)]], "on the boundary of ", Cell[BoxData[ \(TraditionalForm\`D\)]], " which has four parts, ", Cell[BoxData[ \(TraditionalForm\`x\ = \ \(-2\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`x\ = \ 2\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y\ = \ \(-1\)\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`y\ = \ 1\)]], ". We consider each case separaetly." }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`x\ = \ \(\(-2\)\(:\)\)\)], FontWeight->"Bold"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g[y_]\ = \ f[\(-2\), y]\)], "Input"], Cell[BoxData[ \(8 - 8\ y + y\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(g'\)[y]\)], "Input"], Cell[BoxData[ \(\(-8\) + 2\ y\)], "Output"] }, Open ]], Cell[TextData[{ "Since ", Cell[BoxData[ \(TraditionalForm\`g' \((y)\)\ < \ 0\)]], " for ", Cell[BoxData[ \(TraditionalForm\`\(-1\)\ \[LessEqual] \ y\ \[LessEqual] \ 1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`g\)]], " is decreasing on this interval and the extreme values are at the \ endpoints:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g[\(-1\)]\)], "Input"], Cell[BoxData[ \(17\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[1]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`x\ = \ \(\(2\)\(:\)\)\)], FontWeight->"Bold"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g[y_]\ = \ f[2, y]\)], "Input"], Cell[BoxData[ \(8\ y + y\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(g'\)[y]\)], "Input"], Cell[BoxData[ \(8 + 2\ y\)], "Output"] }, Open ]], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`g' \((y)\)\ = \ 0\)]], " for ", Cell[BoxData[ \(TraditionalForm\`y\ = \ \(-4\)\)]], " which is out of range. 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Thus, the extreme values are again at the \ endpoints (which we already computed above):" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g[\(-2\)]\)], "Input"], Cell[BoxData[ \(17\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[2]\)], "Input"], Cell[BoxData[ \(9\)], "Output"] }, Open ]], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`y\ = \ \(\(1\)\(:\)\)\)], FontWeight->"Bold"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g[x_]\ = \ f[x, 1]\)], "Input"], Cell[BoxData[ \(1 + 2\ x + x\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(g'\)[x]\)], "Input"], Cell[BoxData[ \(2 + 2\ x\)], "Output"] }, Open ]], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`g' \((x)\)\ = \ 0\)]], " for ", Cell[BoxData[ \(TraditionalForm\`x\ = \ \(-1\)\)]], " which is in range. 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Metetra .95 .944 .625 r .19192 1.05799 .21959 1.02917 .26333 .94496 .23685 .96504 Metetra .958 .947 .669 r .23685 .96504 .26333 .94496 .30622 .86889 .28078 .88069 Metetra .958 .942 .711 r .28078 .88069 .30622 .86889 .34841 .80057 .32386 .80447 Metetra .949 .928 .748 r .32386 .80447 .34841 .80057 .39006 .73966 .36627 .73599 Metetra .93 .903 .777 r .36627 .73599 .39006 .73966 .43131 .68588 .40814 .6749 Metetra .901 .869 .798 r .40814 .6749 .43131 .68588 .4723 .63899 .44963 .62091 Metetra .864 .828 .81 r .44963 .62091 .4723 .63899 .51316 .5988 .49087 .57379 Metetra .822 .781 .814 r .49087 .57379 .51316 .5988 .55402 .56518 .53201 .53336 Metetra .776 .732 .812 r .53201 .53336 .55402 .56518 .59503 .53803 .57317 .49946 Metetra .73 .682 .804 r .57317 .49946 .59503 .53803 .63632 .51729 .6145 .47201 Metetra .684 .633 .793 r .6145 .47201 .63632 .51729 .67802 .50295 .65613 .45093 Metetra .64 .587 .779 r .65613 .45093 .67802 .50295 .72029 .49504 .69821 .43623 Metetra .6 .544 .764 r .69821 .43623 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.40021 Metetra .676 .611 .775 r .63414 .40021 .65613 .45093 .69821 .43623 .67608 .3788 Metetra .635 .569 .763 r .67608 .3788 .69821 .43623 .74087 .42792 .7185 .36372 Metetra .597 .529 .749 r .7185 .36372 .74087 .42792 .78428 .42607 .76154 .35498 Metetra .966 .926 .601 r .13355 1.12418 .16326 1.0896 .20951 .98754 .18125 1.01259 Metetra .972 .928 .638 r .18125 1.01259 .20951 .98754 .25458 .89459 .22758 .91071 Metetra .973 .924 .673 r .22758 .91071 .25458 .89459 .29865 .81023 .27274 .8179 Metetra .967 .913 .705 r .27274 .8179 .29865 .81023 .3419 .73396 .31692 .73363 Metetra .953 .895 .732 r .31692 .73363 .3419 .73396 .38448 .6654 .36029 .65742 Metetra .932 .87 .754 r .36029 .65742 .38448 .6654 .42654 .6042 .40301 .58887 Metetra .904 .839 .77 r .40301 .58887 .42654 .6042 .46824 .55006 .44523 .52762 Metetra .87 .802 .78 r .44523 .52762 .46824 .55006 .50971 .50277 .4871 .47341 Metetra .832 .761 .784 r .4871 .47341 .50971 .50277 .55109 .46212 .52877 .42598 Metetra .792 .718 .783 r .52877 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(******************************************************************* End of Mathematica Notebook file. *******************************************************************)