(*^ ::[ 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, L0, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L0, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R32768, L0, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L0, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B32768, L0, 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; magnification = 150; currentKernel; ] :[font = subsection; inactive; preserveAspect] Math 225: Calculus III Quiz 5 2/22&2/24 :[font = input; initialization; preserveAspect] *) Needs["Calculus`Advanced`"] (* :[font = subsubsection; inactive; preserveAspect; startGroup] 1. Find the absolute maximum of the function f[x_,y_] = x^2 - 2 x y^2 - 4 y on the rectangle -2 <= x <= 2, -2 <= y <= 2. ;[s] 5:0,0;45,1;76,0;92,1;119,0;122,-1; 2:3,19,14,Times,1,12,0,0,0;2,18,13,Courier,1,12,0,0,0; :[font = subsubsection; inactive; preserveAspect] Solution :[font = input; preserveAspect] f[x_,y_] = x^2 - 2 x y^2 - 4 y; :[font = text; inactive; preserveAspect] First, find critical points inside the region: :[font = input; preserveAspect; startGroup] D[f[x,y],x] == 0 :[font = output; output; inactive; preserveAspect; endGroup] 2*x - 2*y^2 == 0 ;[o] 2 2 x - 2 y == 0 :[font = input; preserveAspect; startGroup] D[f[x,y],y] == 0 :[font = output; output; inactive; preserveAspect; endGroup] -4 - 4*x*y == 0 ;[o] -4 - 4 x y == 0 :[font = text; inactive; preserveAspect] The second eqn says y = -1/x. Plug this into the first eqn to get: ;[s] 3:0,0;20,1;28,0;67,-1; 2:2,19,14,Times,0,12,0,0,0;1,18,13,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] x - (-1/x)^2 == 0 :[font = output; output; inactive; preserveAspect; endGroup] -x^(-2) + x == 0 ;[o] -2 -x + x == 0 :[font = input; preserveAspect] x^3 - 1 == 0 :[font = text; inactive; preserveAspect] So there's only one critical pont: {1,-1}. ;[s] 3:0,0;35,1;41,0;43,-1; 2:2,19,14,Times,0,12,0,0,0;1,18,13,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] f[1,-1] :[font = output; output; inactive; preserveAspect; endGroup] 3 ;[o] 3 :[font = text; inactive; preserveAspect] Check corners: :[font = input; preserveAspect; startGroup] {f[-2,-2], f[-2,2], f[2,-2], f[2,2]} :[font = output; output; inactive; preserveAspect; endGroup] {28, 12, -4, -20} ;[o] {28, 12, -4, -20} :[font = text; inactive; preserveAspect] And critical points on the boundary: :[font = text; inactive; preserveAspect] Left edge: :[font = input; preserveAspect; startGroup] f[-2,y] :[font = output; output; inactive; preserveAspect; endGroup] 4 - 4*y + 4*y^2 ;[o] 2 4 - 4 y + 4 y :[font = input; preserveAspect; startGroup] Solve[D[%,y] == 0] :[font = output; output; inactive; preserveAspect; endGroup] {{y -> 1/2}} ;[o] 1 {{y -> -}} 2 :[font = input; preserveAspect; startGroup] f[-2,1/2] :[font = output; output; inactive; preserveAspect; endGroup] 3 ;[o] 3 :[font = text; inactive; preserveAspect] Right edge: :[font = input; preserveAspect; startGroup] f[2,y] :[font = output; output; inactive; preserveAspect; endGroup] 4 - 4*y - 4*y^2 ;[o] 2 4 - 4 y - 4 y :[font = input; preserveAspect; startGroup] Solve[D[%,y] == 0] :[font = output; output; inactive; preserveAspect; endGroup] {{y -> -1/2}} ;[o] 1 {{y -> -(-)}} 2 :[font = input; preserveAspect; startGroup] f[2,-1/2] :[font = output; output; inactive; preserveAspect; endGroup] 5 ;[o] 5 :[font = text; inactive; preserveAspect] Bottom edge: :[font = input; preserveAspect; startGroup] f[x,-2] :[font = output; output; inactive; preserveAspect; endGroup] 8 - 8*x + x^2 ;[o] 2 8 - 8 x + x :[font = input; preserveAspect; startGroup] Solve[D[%,x] == 0] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 4}} ;[o] {{x -> 4}} :[font = text; inactive; preserveAspect] (Outside of region.) :[font = text; inactive; preserveAspect] Top edge: :[font = input; preserveAspect; startGroup] f[x,2] :[font = output; output; inactive; preserveAspect; endGroup] -8 - 8*x + x^2 ;[o] 2 -8 - 8 x + x :[font = input; preserveAspect; startGroup] Solve[D[%,x] == 0] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 4}} ;[o] {{x -> 4}} :[font = text; inactive; preserveAspect] (Outside of region.) :[font = text; inactive; preserveAspect] Largest value of f found above is f[-2,-2] = 28. ;[s] 5:0,0;17,1;19,0;34,1;47,0;49,-1; 2:3,19,14,Times,0,12,0,0,0;2,18,13,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Plot3D[f[x,y], {x,-2,2},{y,-2,2}, 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.968 r .47584 .53707 .51444 .53756 .53718 .51606 .49827 .51578 Metetra .67 .889 .978 r .49827 .51578 .53718 .51606 .56068 .49558 .52143 .49553 Metetra .644 .895 .984 r .52143 .49553 .56068 .49558 .58501 .47614 .54542 .47633 Metetra .613 .897 .987 r .54542 .47633 .58501 .47614 .61025 .45778 .5703 .45823 Metetra .576 .896 .986 r .5703 .45823 .61025 .45778 .6365 .4405 .59617 .44123 Metetra .534 .889 .979 r .59617 .44123 .6365 .4405 .66385 .42436 .62314 .42539 Metetra .486 .877 .966 r .62314 .42539 .66385 .42436 .69243 .40938 .65132 .41074 Metetra .799 .848 .887 r .29675 .71923 .33462 .71439 .35346 .68605 .31542 .69012 Metetra .79 .854 .898 r .31542 .69012 .35346 .68605 .37267 .65872 .33444 .66202 Metetra .779 .86 .91 r .33444 .66202 .37267 .65872 .39229 .63239 .35385 .63493 Metetra .767 .865 .921 r .35385 .63493 .39229 .63239 .41237 .60705 .37369 .60883 Metetra .753 .87 .933 r .37369 .60883 .41237 .60705 .43296 .58272 .39401 .58374 Metetra .737 .874 .944 r .39401 .58374 .43296 .58272 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.73874 Metetra .882 .924 .847 r .19261 .73874 .23527 .71612 .25416 .68354 .21165 .7023 Metetra .867 .917 .861 r .21165 .7023 .25416 .68354 .27337 .65197 .23094 .66688 Metetra .85 .908 .874 r .23094 .66688 .27337 .65197 .29293 .62139 .25054 .63245 Metetra .831 .896 .886 r .25054 .63245 .29293 .62139 .31289 .59179 .27047 .59901 Metetra .81 .883 .898 r .27047 .59901 .31289 .59179 .33328 .56318 .29079 .56652 Metetra .786 .867 .908 r .29079 .56652 .33328 .56318 .35417 .53553 .31154 .53498 Metetra .76 .848 .917 r .31154 .53498 .35417 .53553 .37561 .50884 .33277 .50437 Metetra .732 .827 .923 r .33277 .50437 .37561 .50884 .39766 .48313 .35453 .47469 Metetra .702 .803 .928 r .35453 .47469 .39766 .48313 .42038 .45838 .37689 .44591 Metetra .67 .777 .93 r .37689 .44591 .42038 .45838 .44385 .43461 .39992 .41804 Metetra .636 .748 .93 r .39992 .41804 .44385 .43461 .46814 .41183 .42369 .39107 Metetra .601 .718 .927 r .42369 .39107 .46814 .41183 .49335 .39005 .44828 .365 Metetra .565 .687 .922 r .44828 .365 .49335 .39005 .51958 .3693 .47378 .33984 Metetra .918 .931 .803 r .1286 .811 .17379 .77623 .19261 .73874 .14781 .76838 Metetra .906 .924 .818 r .14781 .76838 .19261 .73874 .21165 .7023 .16719 .72686 Metetra .891 .914 .833 r .16719 .72686 .21165 .7023 .23094 .66688 .18677 .68639 Metetra .875 .903 .847 r .18677 .68639 .23094 .66688 .25054 .63245 .20657 .64694 Metetra .856 .888 .86 r .20657 .64694 .25054 .63245 .27047 .59901 .22665 .60849 Metetra .834 .871 .871 r .22665 .60849 .27047 .59901 .29079 .56652 .24704 .57099 Metetra .809 .851 .881 r .24704 .57099 .29079 .56652 .31154 .53498 .26779 .53443 Metetra .782 .828 .888 r .26779 .53443 .31154 .53498 .33277 .50437 .28894 .49877 Metetra .752 .802 .893 r .28894 .49877 .33277 .50437 .35453 .47469 .31055 .46399 Metetra .72 .774 .895 r .31055 .46399 .35453 .47469 .37689 .44591 .33267 .43008 Metetra .686 .743 .894 r .33267 .43008 .37689 .44591 .39992 .41804 .35536 .39699 Metetra .651 .711 .891 r .35536 .39699 .39992 .41804 .42369 .39107 .37868 .36473 Metetra .615 .677 .885 r .37868 .36473 .42369 .39107 .44828 .365 .40272 .33326 Metetra .578 .642 .877 r .40272 .33326 .44828 .365 .47378 .33984 .42755 .30258 Metetra .935 .924 .773 r .08061 .85479 .1286 .811 .14781 .76838 .1005 .80563 Metetra .924 .915 .79 r .1005 .80563 .14781 .76838 .16719 .72686 .12048 .75766 Metetra .91 .904 .805 r .12048 .75766 .16719 .72686 .18677 .68639 .14058 .71082 Metetra .894 .89 .819 r .14058 .71082 .18677 .68639 .20657 .64694 .16083 .66507 Metetra .875 .873 .832 r .16083 .66507 .20657 .64694 .22665 .60849 .18127 .62035 Metetra .852 .854 .842 r .18127 .62035 .22665 .60849 .24704 .57099 .20195 .57663 Metetra .827 .831 .851 r .20195 .57663 .24704 .57099 .26779 .53443 .22289 .53386 Metetra .799 .805 .856 r .22289 .53386 .26779 .53443 .28894 .49877 .24415 .49199 Metetra .768 .776 .859 r .24415 .49199 .28894 .49877 .31055 .46399 .26577 .451 Metetra .735 .745 .86 r .26577 .451 .31055 .46399 .33267 .43008 .28781 .41083 Metetra .7 .712 .857 r .28781 .41083 .33267 .43008 .35536 .39699 .31031 .37146 Metetra .664 .677 .852 r .31031 .37146 .35536 .39699 .37868 .36473 .33333 .33284 Metetra .627 .641 .845 r .33333 .33284 .37868 .36473 .40272 .33326 .35695 .29494 Metetra .59 .605 .835 r .35695 .29494 .40272 .33326 .42755 .30258 .38122 .25772 Metetra .948 .911 .748 r .02933 .90854 .08061 .85479 .1005 .80563 .05025 .85123 Metetra .937 .901 .763 r .05025 .85123 .1005 .80563 .12048 .75766 .07116 .79526 Metetra .924 .888 .778 r .07116 .79526 .12048 .75766 .14058 .71082 .09209 .74057 Metetra .907 .873 .792 r .09209 .74057 .14058 .71082 .16083 .66507 .11309 .68709 Metetra .888 .854 .803 r .11309 .68709 .16083 .66507 .18127 .62035 .13418 .63475 Metetra .865 .833 .813 r .13418 .63475 .18127 .62035 .20195 .57663 .1554 .5835 Metetra .84 .808 .82 r .1554 .5835 .20195 .57663 .22289 .53386 .1768 .53327 Metetra .811 .78 .824 r .1768 .53327 .22289 .53386 .24415 .49199 .1984 .48401 Metetra .78 .75 .826 r .1984 .48401 .24415 .49199 .26577 .451 .22027 .43565 Metetra .746 .718 .825 r .22027 .43565 .26577 .451 .28781 .41083 .24243 .38815 Metetra .711 .684 .822 r .24243 .38815 .28781 .41083 .31031 .37146 .26495 .34145 Metetra .675 .648 .816 r .26495 .34145 .31031 .37146 .33333 .33284 .28786 .29548 Metetra .638 .612 .808 r .28786 .29548 .33333 .33284 .35695 .29494 .31124 .2502 Metetra .601 .576 .798 r .31124 .2502 .35695 .29494 .38122 .25772 .33513 .20555 Metetra P p .002 w .34357 .02757 m .89546 .25695 L s .89546 .25695 m .97243 .81054 L s .97243 .81054 m .31536 .62236 L s .31536 .62236 m .34357 .02757 L s .09125 .39965 m .02757 .92301 L s .02757 .92301 m .31536 .62236 L s .31536 .62236 m .34357 .02757 L s .34357 .02757 m .09125 .39965 L s P p p .002 w .09125 .39965 m .34357 .02757 L s P p .002 w .09125 .39965 m .09782 .40174 L s P [(-2)] .07812 .39546 1 .31863 Mshowa p .002 w .1448 .32069 m .15133 .3229 L s P [(-1)] .13174 .31627 1 .33838 Mshowa p .002 w .20402 .23335 m .21051 .23569 L s P [(0)] .19105 .22868 1 .36072 Mshowa p .002 w .26989 .13623 m .27631 .13871 L s P [(1)] .25703 .13126 1 .38623 Mshowa p .002 w .34357 .02757 m .34993 .03022 L s P [(2)] .33084 .02228 1 .41563 Mshowa p .001 w .10155 .38447 m .10548 .38574 L s P p .001 w .11204 .36899 m .11597 .37028 L s P p .001 w .12274 .35321 m .12667 .35451 L s P p .001 w .13366 .33711 m .13758 .33842 L s P p .001 w .15616 .30394 m .16007 .30528 L s P p .001 w .16776 .28683 m .17166 .28819 L s P p .001 w .17959 .26938 m .18349 .27075 L s P p .001 w .19168 .25156 m .19558 .25294 L s P p .001 w .21663 .21476 m .22051 .21618 L s P p .001 w .22951 .19577 m .23339 .1972 L s P p .001 w .24267 .17636 m .24654 .17781 L s P p .001 w .25613 .15651 m .25999 .15799 L s P p .001 w .28395 .11548 m .2878 .11699 L s P p .001 w .29834 .09426 m .30219 .09579 L s P p .001 w .31307 .07255 m .31691 .07409 L s P p .001 w .32814 .05032 m .33197 .05189 L s P P p p .002 w .34357 .02757 m .89546 .25695 L s P p .002 w .34357 .02757 m .3397 .03328 L s P [(-2)] .35131 .01616 -0.67812 1 Mshowa p .002 w .49497 .0905 m .49073 .09593 L s P [(-1)] .50346 .07964 -0.78158 1 Mshowa p .002 w .63686 .14947 m .63229 .15463 L s P [(0)] .646 .13915 -0.88503 1 Mshowa p .002 w .7701 .20485 m .76525 .20975 L s P [(1)] .77979 .19504 -0.98848 1 Mshowa p .002 w .89546 .25695 m .89037 .26161 L s P [(2)] .90562 .24764 -1 .9158 Mshowa p .001 w .37466 .04049 m .37229 .04388 L s P p .001 w .40533 .05324 m .40292 .0566 L s P p .001 w .43561 .06582 m .43315 .06915 L s P p .001 w .46548 .07824 m .46298 .08153 L s P p .001 w .52408 .1026 m .5215 .10582 L s P p .001 w .55282 .11454 m .55019 .11773 L s P p .001 w .58119 .12633 m .57852 .12949 L s P p .001 w .6092 .13797 m .6065 .1411 L s P p .001 w .66418 .16082 m .6614 .16389 L s P p .001 w .69115 .17203 m .68834 .17507 L s P p .001 w .71779 .18311 m .71495 .18611 L s P p .001 w .74411 .19404 m .74123 .19702 L s P p .001 w .79578 .21552 m .79284 .21843 L s P p .001 w .82115 .22607 m .81818 .22895 L s P p .001 w .84622 .23648 m .84322 .23933 L s P p .001 w .87098 .24678 m .86796 .2496 L s P P p p .002 w .89546 .25695 m .97243 .81054 L s P p .002 w .89701 .26813 m .89064 .2655 L s P [(-20)] .90975 .27339 -1 -0.41306 Mshowa p .002 w .92472 .46744 m .91825 .46507 L s P [(0)] .93767 .47219 -1 -0.36706 Mshowa p .002 w .95634 .69483 m .94976 .69277 L s P [(20)] .96949 .69896 -1 -0.31381 Mshowa p .001 w .90228 .306 m .89844 .30445 L s P p .001 w .90768 .34483 m .90383 .34331 L s P p .001 w .91321 .38465 m .90936 .38316 L s P p .001 w .91889 .42551 m .91502 .42405 L s P p .001 w .93071 .51049 m .92682 .5091 L s P p .001 w .93686 .55471 m .93295 .55335 L s P p .001 w .94317 .60014 m .93925 .59882 L s P p .001 w .94967 .64682 m .94573 .64555 L s P p .001 w .96321 .74421 m .95925 .74301 L s P p .001 w .97027 .79501 m .9663 .79386 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] The Unformatted text for this cell was not generated. Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -SurfaceGraphics- :[font = subsubsection; inactive; preserveAspect; startGroup] 2. Find the extreme values of the function f[x_,y_] = x^3 y subject to the constraint x^2 + 4y^2 == 1 ;[s] 4:0,0;44,1;61,0;86,1;103,-1; 2:2,19,14,Times,1,12,0,0,0;2,18,13,Courier,1,12,0,0,0; :[font = subsubsection; inactive; preserveAspect] Solution :[font = input; preserveAspect] f[x_,y_] = x^3 y; :[font = text; inactive; preserveAspect] Must solve simultaneously: :[font = input; preserveAspect; startGroup] D[f[x,y],x] == lambda D[x^2 + 4y^2,x] :[font = output; output; inactive; preserveAspect; endGroup] 3*x^2*y == 2*lambda*x ;[o] 2 3 x y == 2 lambda x :[font = input; preserveAspect; startGroup] D[f[x,y],y] == lambda D[x^2 + 4y^2,y] :[font = output; output; inactive; preserveAspect; endGroup] x^3 == 8*lambda*y ;[o] 3 x == 8 lambda y :[font = text; inactive; preserveAspect] and :[font = input; preserveAspect] x^2 + 4y^2 == 1; :[font = text; inactive; preserveAspect] First, if x == 0, then y = ± 1/2 (from third eqn). Similarly, if y == 0 then x = ±1. These are critical points andf is zero at all of them. To find other critical points, we may assume x ­ 0 and y ­ 0. From the first 2 eqns: ;[s] 15:0,0;10,1;16,0;23,1;32,0;65,1;72,0;77,1;83,0;114,1;115,0;185,1;190,0;195,1;200,0;225,-1; 2:8,19,14,Times,0,12,0,0,0;7,18,13,Courier,1,12,0,0,0; :[font = input; preserveAspect] lambda == (3/2) x y == (1/8) x^3/y; 12 y^2 = x^2; x = ±2Sqrt[3]y :[font = text; inactive; preserveAspect] Plug this into the third eqn: :[font = input; preserveAspect] 12 y^2 + 4 y^2 == 1; y == ±1/4 :[font = text; inactive; preserveAspect] So we get 4 critical points: {±Sqrt[3]/2,±1/4}. ;[s] 2:0,0;29,1;48,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] {f[ Sqrt[3]/2, 1/4], f[-Sqrt[3]/2, 1/4], f[ Sqrt[3]/2,-1/4], f[-Sqrt[3]/2,-1/4]} :[font = output; output; inactive; preserveAspect; endGroup] {3^(3/2)/32, -3^(3/2)/32, -3^(3/2)/32, 3^(3/2)/32} ;[o] 3/2 3/2 3/2 3/2 3 -3 -3 3 {----, -----, -----, ----} 32 32 32 32 :[font = input; preserveAspect; startGroup] N[%] :[font = output; output; inactive; preserveAspect; endGroup] {0.1623797632095822463, -0.1623797632095822463, -0.1623797632095822463, 0.1623797632095822463} ;[o] {0.16238, -0.16238, -0.16238, 0.16238} :[font = text; inactive; preserveAspect] The maximum value is 3^(3/2)/32 = 0.16238 at the points ±{Sqrt[3]/2,1/4}, and the minimum value is -3^(3/2)/32 = -0.16238 at the points ±{Sqrt[3]/2,-1/4}. ;[s] 8:0,0;21,1;41,0;56,1;72,0;99,1;121,0;136,1;155,-1; 2:4,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] Lagrange[f[x,y],x^2 + 4y^2 == 1] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 3^(1/2)/2, y -> 1/4}, {x -> -3^(1/2)/2, y -> 1/4}, {x -> 3^(1/2)/2, y -> -1/4}, {x -> -3^(1/2)/2, y -> -1/4}, {x -> 0, y -> 1/2}, {x -> 0, y -> 1/2}, {x -> 0, y -> -1/2}, {x -> 0, y -> -1/2}} ;[o] Sqrt[3] 1 -Sqrt[3] 1 {{x -> -------, y -> -}, {x -> --------, y -> -}, 2 4 2 4 Sqrt[3] 1 -Sqrt[3] 1 {x -> -------, y -> -(-)}, {x -> --------, y -> -(-)}, 2 4 2 4 1 1 1 {x -> 0, y -> -}, {x -> 0, y -> -}, {x -> 0, y -> -(-)}, 2 2 2 1 {x -> 0, y -> -(-)}} 2 :[font = input; preserveAspect; startGroup] f[x,y] /. % :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {3^(3/2)/32, -3^(3/2)/32, -3^(3/2)/32, 3^(3/2)/32, 0, 0, 0, 0} ;[o] 3/2 3/2 3/2 3/2 3 -3 -3 3 {----, -----, -----, ----, 0, 0, 0, 0} 32 32 32 32 ^*)