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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[ 478267, 9407]*) (*NotebookOutlinePosition[ 478920, 9430]*) (* CellTagsIndexPosition[ 478876, 9426]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["15.7 Maximum and Minimum Values", "Subsection"], 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\ = \ \(x\^2\) y\ - \ 2\ x\ y\ + \ 2 y\^2 - 15 y;\)\)], "Input"], Cell["The first and second partial derivatives are:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[f, x]\)], "Input"], Cell[BoxData[ \(\(-2\)\ y + 2\ x\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f, y]\)], "Input"], Cell[BoxData[ \(\(-15\) - 2\ x + x\^2 + 4\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f, x, x]\)], "Input"], Cell[BoxData[ \(2\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f, y, y]\)], "Input"], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f, x, y]\)], "Input"], Cell[BoxData[ \(\(-2\) + 2\ x\)], "Output"] }, Open ]], Cell["The crititcal points are:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cps\ = \ Solve[{D[f, x]\ \[Equal] \ 0, \ D[f, 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[ \(d\ = \ D[f, x, x] D[f, y, y] - D[f, x, y]^2\)], "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[ \(d\ /. \ cps\)], "Input"], Cell[BoxData[ \({\(-64\), \(-64\), 32}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f, x, x]\ /. \ cps\)], "Input"], Cell[BoxData[ \({0, 0, 8}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f\ /. \ cps\)], "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, \(-5\), 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 def %% SurfaceGraphics 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"Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f, x, y]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["The crititcal points are:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cps\ = \ Solve[{D[f, x]\ \[Equal] \ 0, \ D[f, 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[ \(d\ = \ D[f, x, x] D[f, y, y] - D[f, x, y]^2\)], "Input"], Cell[BoxData[ \(\(-6\)\ \((\(-12\) + 6\ x)\)\)], "Output"] }, Open ]], Cell[TextData[{ "Plug the critical points into ", Cell[BoxData[ \(TraditionalForm\`d\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(d\ /. \ cps\)], "Input"], Cell[BoxData[ \({72, \(-72\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f, x, x]\ /. \ cps\)], "Input"], Cell[BoxData[ \({\(-12\), 12}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f\ /. \ cps\)], "Input"], Cell[BoxData[ \({0, \(-32\)}\)], "Output"] }, Open ]], Cell[TextData[{ "Since ", Cell[BoxData[ \(TraditionalForm\`d(0, 0)\ = 72\ > \ 0\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`\(f\_xx\)(0, 0)\ = \(-\ 12\)\ < \ 0\)]], ", the graph of ", Cell[BoxData[ \(TraditionalForm\`f\)]], " curves down in all directions at this point and thus ", Cell[BoxData[ \(TraditionalForm\`f\)]], " has a local maximum at ", Cell[BoxData[ \(TraditionalForm\`\((0, 0)\)\)]], ".\nSince ", Cell[BoxData[ \(TraditionalForm\`d(4, 0) = \ \(-72\) < 0\)]], ", the graph of ", 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, \(-3\), 7}, {y, \(-5\), 5}, \ BoxRatios \[Rule] {1, 1, 2}]\)], "Input"], Cell[GraphicsData["PostScript", 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.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 .43841 .61679 Metetra .689 .535 .671 r .43841 .61679 .46192 .71422 .50584 .66873 .48219 .57155 Metetra .629 .497 .682 r .48219 .57155 .50584 .66873 .54983 .64559 .52594 .54824 Metetra .501 .414 .69 r .52594 .54824 .54983 .64559 .59452 .65604 .57026 .55782 Metetra .237 .24 .659 r .57026 .55782 .59452 .65604 .64093 .71267 .6161 .61265 Metetra 0 0 .503 r .6161 .61265 .64093 .71267 .69057 .83119 .66489 .72807 Metetra 0 0 .237 r .66489 .72807 .69057 .83119 .74568 1.03324 .71876 .92522 Metetra .748 .5 0 r .10085 .35794 .12033 .4605 .1487 .62355 .12838 .5165 Metetra 0 0 .432 r .12838 .5165 .1487 .62355 .18269 .73117 .16167 .62086 Metetra .234 .21 .627 r .16167 .62086 .18269 .73117 .22108 .78701 .19949 .67464 Metetra .496 .391 .665 r .19949 .67464 .22108 .78701 .26254 .79853 .24051 .68515 Metetra .617 .472 .662 r .24051 .68515 .26254 .79853 .30586 .77575 .28349 .6622 Metetra .673 .509 .655 r .28349 .6622 .30586 .77575 .35003 .73009 .32739 .61693 Metetra .696 .524 .651 r .32739 .61693 .35003 .73009 .39434 .67329 .37149 .56079 Metetra .696 .524 .651 r .37149 .56079 .39434 .67329 .43841 .61679 .41534 .50496 Metetra .673 .509 .655 r .41534 .50496 .43841 .61679 .48219 .57155 .45886 .4601 Metetra .618 .473 .663 r .45886 .4601 .48219 .57155 .52594 .54824 .50227 .43662 Metetra .507 .4 .667 r .50227 .43662 .52594 .54824 .57026 .55782 .5461 .44521 Metetra .289 .254 .645 r .5461 .44521 .57026 .55782 .6161 .61265 .59125 .49788 Metetra 0 0 .533 r .59125 .49788 .6161 .61265 .66489 .72807 .63907 .60955 Metetra 0 0 .309 r .63907 .60955 .66489 .72807 .71876 .92522 .69156 .8007 Metetra 0 0 .161 r .08261 .2433 .10085 .35794 .12838 .5165 .10929 .39664 Metetra 0 0 .473 r .10929 .39664 .12838 .5165 .16167 .62086 .14184 .49722 Metetra .282 .227 .616 r .14184 .49722 .16167 .62086 .19949 .67464 .17902 .54863 Metetra .502 .38 .646 r .17902 .54863 .19949 .67464 .24051 .68515 .2195 .55801 Metetra .608 .453 .647 r .2195 .55801 .24051 .68515 .28349 .6622 .26204 .53491 Metetra .659 .489 .642 r .26204 .53491 .28349 .6622 .32739 .61693 .30558 .49015 Metetra .681 .503 .639 r .30558 .49015 .32739 .61693 .37149 .56079 .34934 .43485 Metetra .681 .503 .639 r .34934 .43485 .37149 .56079 .41534 .50496 .39286 .37984 Metetra .659 .489 .642 r .39286 .37984 .41534 .50496 .45886 .4601 .43602 .33548 Metetra .609 .455 .647 r .43602 .33548 .45886 .4601 .50227 .43662 .47898 .31187 Metetra .51 .388 .65 r .47898 .31187 .50227 .43662 .5461 .44521 .52223 .31935 Metetra .326 .263 .631 r .52223 .31935 .5461 .44521 .59125 .49788 .56659 .36954 Metetra .011 .045 .549 r .56659 .36954 .59125 .49788 .63907 .60955 .61333 .47686 Metetra 0 0 .365 r .61333 .47686 .63907 .60955 .69156 .8007 .66431 .66104 Metetra 0 g .25 Mabswid .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 0 1.32748 m .37669 1.62453 L s .04162 1.3603 m .04233 1.35011 L s [(-4)] .0402 1.38069 .06968 -1 Mshowa .12202 1.4237 m .1226 1.4135 L s [(-2)] .12085 1.44411 .05726 -1 Mshowa .19883 1.48428 m .1993 1.47407 L s [(0)] .1979 1.5047 .04561 -1 Mshowa .2723 1.54221 m .27266 1.532 L s [(2)] .27159 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 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variables."\)], \ "Message"], Cell[BoxData[ \({{x \[Rule] y}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(d\ = \ D[f, x, x] D[f, y, y] - D[f, x, y]^2\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f\ /. \ cps\)], "Input"], Cell[BoxData[ \({0}\)], "Output"] }, Open ]], Cell[TextData[{ "The Second Derivative Test is inconclusive since ", Cell[BoxData[ \(TraditionalForm\`d\)]], " is always 0. There is a whole line of critical points, ", Cell[BoxData[ \(TraditionalForm\`y\ = \ x\)]], ", where the function has the value 0. 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"Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[f, x]\)], "Input"], Cell[BoxData[ \(\(-2\) + 2\ x + 4\ y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(D[f, y]\)], "Input"], Cell[BoxData[ \(4\ x + 2\ y\)], "Output"] }, Open ]], Cell["The crititcal points are:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(cps\ = \ Solve[{D[f, x]\ \[Equal] \ 0, \ D[f, 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[ \(f\ /. \ cps\)], "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[ \(\((\(-2\)\ x + x\^2 + 4\ x\ y + y\^2)\)[\(-2\), y]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(g'\)[y]\)], "Input"], Cell[BoxData[ RowBox[{ SuperscriptBox[\((\(-2\)\ x + x\^2 + 4\ x\ y + y\^2)\), TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(\(-2\), 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[ \(\((1 - 6\ x + x\^2)\)[\(-2\), \(-1\)]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[1]\)], "Input"], Cell[BoxData[ \(\((1 + 2\ x + x\^2)\)[\(-2\), 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[ \(\((\(-2\)\ x + x\^2 + 4\ x\ y + y\^2)\)[2, y]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(g'\)[y]\)], "Input"], Cell[BoxData[ RowBox[{ SuperscriptBox[\((\(-2\)\ x + x\^2 + 4\ x\ y + y\^2)\), TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(2, y\), "]"}]], "Output"] }, Open ]], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`g' \((y)\)\ = \ 0\)]], " for ", Cell[BoxData[ \(TraditionalForm\`y\ = \ \(-4\)\)]], " which is out of range. Thus, the extreme values are again at the \ endpoints:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g[\(-1\)]\)], "Input"], Cell[BoxData[ \(\((1 - 6\ x + x\^2)\)[2, \(-1\)]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[1]\)], "Input"], Cell[BoxData[ \(\((1 + 2\ x + x\^2)\)[2, 1]\)], "Output"] }, Open ]], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`y\ = \ \(\(-1\)\(:\)\)\)], FontWeight->"Bold"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g[x_]\ = \ f[x, \(-1\)]\)], "Input"], Cell[BoxData[ \(\((\(-2\)\ x + x\^2 + 4\ x\ y + y\^2)\)[x, \(-1\)]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(g'\)[x]\)], "Input"], Cell[BoxData[ RowBox[{ SuperscriptBox[\((\(-2\)\ x + x\^2 + 4\ x\ y + y\^2)\), TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, \(-1\)\), "]"}]], "Output"] }, Open ]], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`g' \((x)\)\ = \ 0\)]], " for ", Cell[BoxData[ \(TraditionalForm\`x\ = \ 3\)]], " which is out of range. Thus, the extreme values are again at the \ endpoints (which we already computed above):" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g[\(-2\)]\)], "Input"], Cell[BoxData[ \(\((8 - 8\ y + y\^2)\)[\(-2\), \(-1\)]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[2]\)], "Input"], Cell[BoxData[ \(\((8\ y + y\^2)\)[2, \(-1\)]\)], "Output"] }, Open ]], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`y\ = \ \(\(1\)\(:\)\)\)], FontWeight->"Bold"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g[x_]\ = \ f[x, 1]\)], "Input"], Cell[BoxData[ \(\((\(-2\)\ x + x\^2 + 4\ x\ y + y\^2)\)[x, 1]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(g'\)[x]\)], "Input"], Cell[BoxData[ RowBox[{ SuperscriptBox[\((\(-2\)\ x + x\^2 + 4\ x\ y + y\^2)\), TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, 1\), "]"}]], "Output"] }, Open ]], Cell[TextData[{ " ", Cell[BoxData[ \(TraditionalForm\`g' \((x)\)\ = \ 0\)]], " for ", Cell[BoxData[ \(TraditionalForm\`x\ = \ \(-1\)\)]], " which is in range. Thus, the extreme values of ", Cell[BoxData[ \(TraditionalForm\`g\)]], " are at the endpoints (which we already computed above) or at this \ critical point:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g[\(-2\)]\)], "Input"], Cell[BoxData[ \(\((8 - 8\ y + y\^2)\)[\(-2\), 1]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[2]\)], "Input"], Cell[BoxData[ \(\((8\ y + y\^2)\)[2, 1]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[\(-1\)]\)], "Input"], Cell[BoxData[ \(\((3 - 4\ y + y\^2)\)[\(-1\), 1]\)], "Output"] }, Open ]], Cell[TextData[{ "Considering all the critical points found in the above steps, the maximum \ of ", Cell[BoxData[ \(TraditionalForm\`f(\(-2\), \(-1\))\ = \ 17\)]], " is 17 and the minimum is ", Cell[BoxData[ \(TraditionalForm\`f(2, \(-1\)) = \(-7\)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot3D[f, {x, \(-2\), 2}, {y, \(-1\), 1}, \ BoxRatios \[Rule] {1, 1, 2}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! 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