(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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[ 793243, 31826]*) (*NotebookOutlinePosition[ 794373, 31862]*) (* CellTagsIndexPosition[ 794329, 31858]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["Maxima & Minima"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "This notebook shows how to solve for critical points and extreme values of a \ function of several variables."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Needs[\"Calculus`Advanced`\"]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Example:"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[" Classify the critical points of"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["f[x_,y_] = x^3 - 4 x y + y^2 + x^2;"], "Input", AspectRatioFixed->True], Cell[TextData["Solution:"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["cp = CriticalPoints[f[x,y]]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{x -> 2, y -> 4}, {x -> 0, y -> 0}}\ \>", "\<\ {{x -> 2, y -> 4}, {x -> 0, y -> 0}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["d = Discriminant[f[x,y]]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 12*(-1 + x)\ \>", "\<\ 12 (-1 + x)\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "Evaluate ", StyleBox["d", FontFamily->"Courier", FontWeight->"Bold"], " at the critical points:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["d /. cp"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {12, -12}\ \>", "\<\ {12, -12}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "So", StyleBox[" {0,0} ", FontFamily->"Courier", FontWeight->"Bold"], "is a saddle point. 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