(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; currentKernel; ] :[font = subsection; inactive; preserveAspect] Math 225: Calculus III Quiz 6 3/5&7/96 :[font = input; initialization; preserveAspect] *) < 0, x -> 0}, {y -> 0, x -> 1}, {y -> 0, x -> 2}, {y -> -2^(1/2), x -> 1 - I}, {y -> -2^(1/2), x -> 1 + I}, {y -> 2^(1/2), x -> 1 - I}, {y -> 2^(1/2), x -> 1 + I}} ;[o] {{y -> 0, x -> 0}, {y -> 0, x -> 1}, {y -> 0, x -> 2}, {y -> -Sqrt[2], x -> 1 - I}, {y -> -Sqrt[2], x -> 1 + I}, {y -> Sqrt[2], x -> 1 - I}, {y -> Sqrt[2], x -> 1 + I}} :[font = text; inactive; preserveAspect] In one step: :[font = input; preserveAspect; startGroup] CriticalPoints[f[x,y]] :[font = output; output; inactive; preserveAspect; endGroup] {{y -> 0, x -> 0}, {y -> 0, x -> 1}, {y -> 0, x -> 2}, {y -> -2^(1/2), x -> 1 - I}, {y -> -2^(1/2), x -> 1 + I}, {y -> 2^(1/2), x -> 1 - I}, {y -> 2^(1/2), x -> 1 + I}} ;[o] {{y -> 0, x -> 0}, {y -> 0, x -> 1}, {y -> 0, x -> 2}, {y -> -Sqrt[2], x -> 1 - I}, {y -> -Sqrt[2], x -> 1 + I}, {y -> Sqrt[2], x -> 1 - I}, {y -> Sqrt[2], x -> 1 + I}} :[font = text; inactive; preserveAspect] We must ignore the complex solutions (the last four). Now we compute the discriminant. :[font = input; preserveAspect] m = {{D[f[x,y],x,x],D[f[x,y],x,y]}, {D[f[x,y],y,x],D[f[x,y],y,y]}}; :[font = input; preserveAspect; startGroup] m //MatrixForm :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{12*(-1 + x)^2 + 4*(-1 + y^2), 8*(-1 + x)*y}, {8*(-1 + x)*y, 4 + 4*(-1 + x)^2}}] ;[o] 2 2 12 (-1 + x) + 4 (-1 + y ) 8 (-1 + x) y 2 8 (-1 + x) y 4 + 4 (-1 + x) :[font = input; preserveAspect; startGroup] d = Det[m] //Simplify :[font = output; output; inactive; preserveAspect; endGroup] 16*(2 - 6*x + 3*x^2)*(2 - 2*x + x^2 - y^2) ;[o] 2 2 2 16 (2 - 6 x + 3 x ) (2 - 2 x + x - y ) :[font = text; inactive; preserveAspect] In one step: :[font = input; preserveAspect; startGroup] d = Discriminant[f[x,y]] :[font = output; output; inactive; preserveAspect; endGroup] 16*(2 - 6*x + 3*x^2)*(2 - 2*x + x^2 - y^2) ;[o] 2 2 2 16 (2 - 6 x + 3 x ) (2 - 2 x + x - y ) :[font = input; preserveAspect; startGroup] d /. cp :[font = output; output; inactive; preserveAspect; endGroup] {64, -16, 64, 128, 128, 128, 128} ;[o] {64, -16, 64, 128, 128, 128, 128} :[font = input; preserveAspect; startGroup] D[f[x,y],x,x] /. cp :[font = output; output; inactive; preserveAspect; endGroup] {8, -4, 8, -8, -8, -8, -8} ;[o] {8, -4, 8, -8, -8, -8, -8} :[font = text; inactive; preserveAspect] Since d < 0 at (1,0), f[x,y] has a saddle point at (1,0). 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.33263 .6042 .3595 .64051 .3304 .62158 .3026 Metetra .66 .698 .874 r .62158 .3026 .64051 .3304 .67744 .31906 .65864 .29006 Metetra 0 .283 .785 r .65864 .29006 .67744 .31906 .71631 .3501 .69747 .31962 Metetra .829 .425 0 r .69747 .31962 .71631 .3501 .75965 .45888 .7405 .42656 Metetra .943 .589 .193 r .7405 .42656 .75965 .45888 .81216 .70109 .79223 .66641 Metetra .95 .731 .57 r .22442 .83038 .24748 .84233 .29416 .58231 .27249 .56659 Metetra .938 .755 .629 r .27249 .56659 .29416 .58231 .33372 .45107 .31283 .43295 Metetra .886 .788 .748 r .31283 .43295 .33372 .45107 .37001 .39565 .34953 .37588 Metetra .695 .773 .911 r .34953 .37588 .37001 .39565 .40518 .38179 .38492 .36085 Metetra .42 .655 .957 r .38492 .36085 .40518 .38179 .44035 .38534 .42023 .36355 Metetra .415 .644 .952 r .42023 .36355 .44035 .38534 .47601 .38922 .45602 .36687 Metetra .592 .719 .932 r .45602 .36687 .47601 .38922 .51216 .38294 .49234 .36029 Metetra .732 .764 .878 r .49234 .36029 .51216 .38294 .54859 .36291 .52898 .3402 Metetra .794 .777 .837 r .52898 .3402 .54859 .36291 .58506 .33263 .56572 .31012 Metetra .793 .781 .842 r .56572 .31012 .58506 .33263 .62158 .3026 .60251 .28057 Metetra .652 .75 .921 r .60251 .28057 .62158 .3026 .65864 .29006 .6398 .26886 Metetra 0 .182 .688 r .6398 .26886 .65864 .29006 .69747 .31962 .67875 .29972 Metetra .882 .465 .049 r .67875 .29972 .69747 .31962 .7405 .42656 .72171 .40872 Metetra .948 .591 .246 r .72171 .40872 .7405 .42656 .79223 .66641 .77304 .65206 Metetra .969 .754 .559 r .19983 .8387 .22442 .83038 .27249 .56659 .24976 .56537 Metetra .965 .791 .618 r .24976 .56537 .27249 .56659 .31283 .43295 .29116 .42545 Metetra .927 .851 .748 r .29116 .42545 .31283 .43295 .34953 .37588 .32846 .36395 Metetra .728 .868 .947 r .32846 .36395 .34953 .37588 .38492 .36085 .36418 .34571 Metetra .394 .716 .987 r .36418 .34571 .38492 .36085 .42023 .36355 .39971 .34615 Metetra .388 .686 .978 r .39971 .34615 .42023 .36355 .45602 .36687 .43567 .34807 Metetra .585 .759 .961 r .43567 .34807 .45602 .36687 .49234 .36029 .47219 .3409 Metetra .735 .798 .903 r .47219 .3409 .49234 .36029 .52898 .3402 .50908 .32103 Metetra .8 .812 .861 r .50908 .32103 .52898 .3402 .56572 .31012 .54611 .292 Metetra .795 .824 .875 r .54611 .292 .56572 .31012 .60251 .28057 .58323 .26437 Metetra .594 .797 .977 r .58323 .26437 .60251 .28057 .6398 .26886 .62083 .25554 Metetra .546 .003 0 r .62083 .25554 .6398 .26886 .67875 .29972 .66003 .29054 Metetra .903 .489 .15 r .66003 .29054 .67875 .29972 .72171 .40872 .70311 .4056 Metetra .945 .589 .29 r .70311 .4056 .72171 .40872 .77304 .65206 .75438 .65848 Metetra .984 .774 .542 r .17338 .86819 .19983 .8387 .24976 .56537 .22574 .57913 Metetra .985 .82 .597 r .22574 .57913 .24976 .56537 .29116 .42545 .26855 .42884 Metetra .958 .903 .724 r .26855 .42884 .29116 .42545 .32846 .36395 .30667 .36001 Metetra .763 .954 .936 r .30667 .36001 .32846 .36395 .36418 .34571 .34289 .33647 Metetra .389 .785 .987 r .34289 .33647 .36418 .34571 .39971 .34615 .37873 .33321 Metetra .371 .735 .99 r .37873 .33321 .39971 .34615 .43567 .34807 .41492 .33284 Metetra .577 .801 .984 r .41492 .33284 .43567 .34807 .47219 .3409 .45167 .3248 Metetra .732 .835 .929 r .45167 .3248 .47219 .3409 .50908 .32103 .48883 .30545 Metetra .796 .85 .891 r .48883 .30545 .50908 .32103 .54611 .292 .52618 .27834 Metetra .777 .872 .918 r .52618 .27834 .54611 .292 .58323 .26437 .56366 .25409 Metetra .412 .79 .992 r .56366 .25409 .58323 .26437 .62083 .25554 .60164 .25027 Metetra .714 .18 0 r .60164 .25027 .62083 .25554 .66003 .29054 .6412 .29236 Metetra .904 .503 .227 r .6412 .29236 .66003 .29054 .70311 .4056 .68457 .41769 Metetra .937 .585 .326 r .68457 .41769 .70311 .4056 .75438 .65848 .73607 .68659 Metetra .994 .79 .523 r .1447 .92025 .17338 .86819 .22574 .57913 .20019 .60863 Metetra .996 .84 .567 r .20019 .60863 .22574 .57913 .26855 .42884 .24482 .44355 Metetra .971 .929 .673 r .24482 .44355 .26855 .42884 .30667 .36001 .28405 .36432 Metetra .785 .996 .862 r .28405 .36432 .30667 .36001 .34289 .33647 .32094 .33328 Metetra .412 .847 .941 r .32094 .33328 .34289 .33647 .37873 .33321 .35721 .3248 Metetra .37 .786 .978 r .35721 .3248 .37873 .33321 .41492 .33284 .39371 .32127 Metetra .565 .842 .997 r .39371 .32127 .41492 .33284 .45167 .3248 .43073 .31205 Metetra .718 .873 .954 r .43073 .31205 .45167 .3248 .48883 .30545 .46818 .29351 Metetra .776 .891 .926 r .46818 .29351 .48883 .30545 .52618 .27834 .50587 .26923 Metetra .712 .915 .964 r .50587 .26923 .52618 .27834 .56366 .25409 .54373 .24988 Metetra .01 .576 .789 r .54373 .24988 .56366 .25409 .60164 .25027 .58213 .25329 Metetra .785 .295 .05 r .58213 .25329 .60164 .25027 .6412 .29236 .62211 .30561 Metetra .899 .512 .283 r .62211 .30561 .6412 .29236 .68457 .41769 .66592 .44579 Metetra .928 .581 .354 r .66592 .44579 .68457 .41769 .73607 .68659 .7179 .7379 Metetra .999 .8 .501 r .11332 .99687 .1447 .92025 .20019 .60863 .17279 .65494 Metetra .998 .849 .531 r .17279 .65494 .20019 .60863 .24482 .44355 .21978 .47017 Metetra .966 .928 .607 r .21978 .47017 .24482 .44355 .28405 .36432 .26046 .37721 Metetra .795 .991 .748 r .26046 .37721 .28405 .36432 .32094 .33328 .29824 .33634 Metetra .462 .889 .856 r .29824 .33634 .32094 .33328 .35721 .3248 .33508 .32106 Metetra .387 .832 .939 r .33508 .32106 .35721 .3248 .39371 .32127 .37198 .31342 Metetra .548 .876 .992 r .37198 .31342 .39371 .32127 .43073 .31205 .40931 .30271 Metetra .687 .906 .973 r .40931 .30271 .43073 .31205 .46818 .29351 .44707 .28531 Metetra .722 .925 .96 r .44707 .28531 .46818 .29351 .50587 .26923 .4851 .26479 Metetra .531 .9 .964 r .4851 .26479 .50587 .26923 .54373 .24988 .52335 .25194 Metetra .402 0 0 r .52335 .25194 .54373 .24988 .58213 .25329 .56219 .26494 Metetra .814 .366 .175 r .56219 .26494 .58213 .25329 .62211 .30561 .60265 .33091 Metetra .892 .519 .324 r .60265 .33091 .62211 .30561 .66592 .44579 .64699 .49101 Metetra .919 .578 .375 r .64699 .49101 .66592 .44579 .7179 .7379 .69967 .81457 Metetra 1 .806 .479 r .07865 1.10085 .11332 .99687 .17279 .65494 .1432 .71949 Metetra .993 .849 .496 r .1432 .71949 .17279 .65494 .21978 .47017 .1932 .50949 Metetra .954 .911 .543 r .1932 .50949 .21978 .47017 .26046 .37721 .23573 .39911 Metetra .804 .962 .639 r .23573 .39911 .26046 .37721 .29824 .33634 .27468 .34589 Metetra .53 .91 .757 r .27468 .34589 .29824 .33634 .33508 .32106 .31226 .32212 Metetra .423 .868 .878 r .31226 .32212 .33508 .32106 .37198 .31342 .34965 .30941 Metetra .525 .898 .963 r .34965 .30941 .37198 .31342 .40931 .30271 .38735 .29689 Metetra .625 .923 .975 r .38735 .29689 .40931 .30271 .44707 .28531 .42544 .28093 Metetra .597 .926 .965 r .42544 .28093 .44707 .28531 .4851 .26479 .4638 .26516 Metetra .148 .692 .778 r .4638 .26516 .4851 .26479 .52335 .25194 .50243 .26051 Metetra .617 .081 0 r .50243 .26051 .52335 .25194 .56219 .26494 .54169 .28568 Metetra .829 .413 .253 r .54169 .28568 .56219 .26494 .60265 .33091 .58265 .36907 Metetra .887 .525 .354 r .58265 .36907 .60265 .33091 .64699 .49101 .62759 .55488 Metetra .912 .575 .391 r .62759 .55488 .64699 .49101 .69967 .81457 .68115 .91964 Metetra .998 .808 .458 r .04 1.23594 .07865 1.10085 .1432 .71949 .11101 .80419 Metetra .986 .843 .464 r .11101 .80419 .1432 .71949 .1932 .50949 .16481 .56251 Metetra .943 .891 .49 r .16481 .56251 .1932 .50949 .23573 .39911 .20969 .43058 Metetra .82 .931 .555 r .20969 .43058 .23573 .39911 .27468 .34589 .25013 .36224 Metetra .605 .917 .666 r .25013 .36224 .27468 .34589 .31226 .32212 .28865 .32818 Metetra .473 .892 .804 r .28865 .32818 .31226 .32212 .34965 .30941 .32666 .30935 Metetra .496 .903 .91 r .32666 .30935 .34965 .30941 .38735 .29689 .36479 .29469 Metetra .521 .907 .94 r .36479 .29469 .38735 .29689 .42544 .28093 .40321 .28052 Metetra .351 .825 .875 r .40321 .28052 .42544 .28093 .4638 .26516 .44189 .27054 Metetra .276 0 0 r .44189 .27054 .4638 .26516 .50243 .26051 .48087 .27592 Metetra .716 .238 .11 r .48087 .27592 .50243 .26051 .54169 .28568 .52052 .31608 Metetra .839 .447 .305 r .52052 .31608 .54169 .28568 .58265 .36907 .56195 .42115 Metetra .884 .532 .376 r .56195 .42115 .58265 .36907 .62759 .55488 .6075 .63943 Metetra .907 .574 .404 r .6075 .63943 .62759 .55488 .68115 .91964 .66207 1.05722 Metetra P p .002 w .63667 .03779 m .89148 .39898 L s .89148 .39898 m .96221 1.39843 L s .96221 1.39843 m .66282 1.08731 L s .66282 1.08731 m .63667 .03779 L s .11572 .24227 m .03779 1.26494 L s .03779 1.26494 m .66282 1.08731 L s .66282 1.08731 m .63667 .03779 L s .63667 .03779 m .11572 .24227 L s P p p .002 w .03779 1.26494 m .38601 1.53954 L s P p .002 w .03779 1.26494 m .03851 1.25552 L s P [(-1)] .03635 1.28378 .0762 -1 Mshowa p .002 w .13232 1.33949 m .13289 1.33006 L s P [(-0.5)] .13119 1.35835 .06029 -1 Mshowa p .002 w .2216 1.40989 m .22203 1.40045 L s P [(0)] .22073 1.42876 .04561 -1 Mshowa p .002 w .30603 1.47647 m .30633 1.46703 L s P [(0.5)] .30543 1.49536 .03202 -1 Mshowa p .002 w .38601 1.53954 m .38619 1.5301 L s P [(1)] .38564 1.55843 .01941 -1 Mshowa p .001 w .05714 1.2802 m .05755 1.27455 L s P p .001 w .07626 1.29528 m .07666 1.28963 L s P p .001 w .09517 1.31019 m .09554 1.30453 L s P p .001 w .11385 1.32492 m .11421 1.31927 L s P p .001 w .15059 1.35389 m .15091 1.34823 L s P p .001 w .16864 1.36813 m .16895 1.36247 L s P p .001 w .18649 1.3822 m .18678 1.37654 L s P p .001 w .20414 1.39612 m .20442 1.39046 L s P p .001 w .23886 1.4235 m .2391 1.41784 L s P p .001 w .25593 1.43696 m .25615 1.4313 L s P p .001 w .27281 1.45028 m .27302 1.44461 L s P p .001 w .28951 1.46344 m .28971 1.45778 L s P p .001 w .32237 1.48936 m .32254 1.48369 L s P p .001 w .33854 1.50211 m .33869 1.49644 L s P p .001 w .35453 1.51472 m .35467 1.50905 L s P p .001 w .37035 1.5272 m .37048 1.52153 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] SurfaceGraphics["<<>>"] ;[o] -SurfaceGraphics- :[font = subsubsection; inactive; preserveAspect; startGroup] 2. Find the maximum and minimum values of the function f[x_,y_] = 3x - 2y subject to the constraint 4x^2 + y^2 == 4 ;[s] 4:0,0;56,1;75,0;101,1;117,-1; 2:2,13,9,Times,1,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = subsubsection; inactive; preserveAspect; endGroup] Solution :[font = input; preserveAspect] f[x_,y_] = 3x - 2y; :[font = input; preserveAspect] g[x_,y_] = 4x^2 + y^2; :[font = input; preserveAspect; startGroup] eqns = {D[f[x,y],x] == lambda D[g[x,y],x], D[f[x,y],y] == lambda D[g[x,y],y], g[x,y] == 4} :[font = output; output; inactive; preserveAspect; endGroup] {3 == 8*lambda*x, -2 == 2*lambda*y, 4*x^2 + y^2 == 4} ;[o] 2 2 {3 == 8 lambda x, -2 == 2 lambda y, 4 x + y == 4} :[font = text; inactive; preserveAspect] From the first two equations we see that neither x nor y can be zero, so we can divide to get lambda == 3/(8x) = -1/y y == (8/3) x ;[s] 6:0,0;49,1;50,0;55,1;56,0;94,1;139,-1; 2:3,13,9,Times,0,12,0,0,0;3,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] Plug this into the last equation: :[font = input; preserveAspect] 4x^2 + (8x/3)^2 == 4 (4 + 64/9)x^2 == 4 x^2 == 4(9/100) == 9/25 :[font = text; inactive; preserveAspect] Thus x == ± 3/5 and y == ± 8/5, and the critical points are (3/5, 8/5) and (-3/5, -8/5). Using Mathematica to solve: ;[s] 10:0,0;5,1;15,0;20,1;30,0;60,1;70,0;75,1;87,0;116,1;117,-1; 2:5,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] lp = Solve[eqns] :[font = output; output; inactive; preserveAspect; endGroup] {{lambda -> -5/8, x -> -3/5, y -> 8/5}, {lambda -> 5/8, x -> 3/5, y -> -8/5}} ;[o] 5 3 8 {{lambda -> -(-), x -> -(-), y -> -}, 8 5 5 5 3 8 {lambda -> -, x -> -, y -> -(-)}} 8 5 5 :[font = text; inactive; preserveAspect] In one step: :[font = input; preserveAspect; startGroup] lp = Lagrange[f[x,y],4x^2 + y^2 == 4] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> -3/5, y -> 8/5}, {x -> 3/5, y -> -8/5}} ;[o] 3 8 3 8 {{x -> -(-), y -> -}, {x -> -, y -> -(-)}} 5 5 5 5 :[font = text; inactive; preserveAspect] Evaluate the function at thee points to see which i sthe maximum and which is the minimum. :[font = input; preserveAspect; startGroup] f[x,y] /. lp :[font = output; output; inactive; preserveAspect; endGroup] {-5, 5} ;[o] {-5, 5} :[font = text; inactive; preserveAspect] Therefore, f[x,y] has a maximum of 5 at the point (3/5, 8/5), and a minimum of -5 at the point (-3/5,-8/5). ;[s] 10:0,0;11,1;17,0;35,1;36,0;50,1;60,0;79,1;81,0;95,1;108,-1; 2:5,13,9,Times,0,12,0,0,0;5,13,10,Courier,1,12,0,0,0; ^*)