%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Wednesday, February 10, 1999 13:29:48} %TCIDATA{LastRevised=Sunday, February 13, 2005 16:58:03} %TCIDATA{} %TCIDATA{} %TCIDATA{Language=American English} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za II online - Maximum a minimum re\U{e1}lnej funkcie viacer\U{fd}ch premenn\U{fd}ch - Extr\U{e9}my a stacin\U{e1}rne body\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Maximum a minimum re\'{a}lnej funkcie viacer\'{y}ch premenn\'{y}ch} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma4.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma42.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma42.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O4.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C4.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \paragraph{Ciele} Po pre\v{s}tudovan\'{\i} tejto \v{c}asti by ste mali by\v{t} schopn\'{\i}: \begin{itemize} \item vysvetli\v{t} pojem lok\'{a}lneho minima, alebo maxima funkcie, \item vypo\v{c}\'{\i}ta\v{t} stacion\'{a}rne body funkcie viacer\'{y}ch premenn\'{y}ch a ur\v{c}i\v{t} relat\'{\i}vne extr\'{e}my, \item vysvetli\v{t} a aplikova\v{t} nutn\'{u} a posta\v{c}uj\'{u}cu podmienku existencie extr\'{e}mu pri konkr\'{e}tnych v\'{y}po\v{c}toch, \item vysvetli\v{t} pojem funkcie ur\v{c}enej implicitne, \item vypo\v{c}\'{\i}ta\v{t} viazan\'{e} extr\'{e}my funkcie, \item vysvetli\v{t} podmienky existencie lok\'{a}lnych extr\'{e}mov, \item n\'{a}js\v{t} absol\'{u}tne extr\'{e}my diferencovate\v{l}nej funkcie na kompaktnej mno\v{z}ine. \end{itemize} \paragraph{Po\v{z}adovan\'{e} vedomosti:} \begin{itemize} \item znalos\v{t} parci\'{a}lnych deriv\'{a}ci\'{\i} r\^{o}znych r\'{a}dov, ich vlastnost\'{\i} a v\'{y}po\v{c}tov, \item znalos\v{t} element\'{a}rnych funkci\'{\i} a ich z\'{a}kladn\'{y}ch vlastnost\'{\i}. \end{itemize} \subsection{Extr\'{e}my a stacion\'{a}rne body.} V tejto kapitole sa budeme zaobera\v{t} iba skal\'{a}rnymi po\v{l}ami, t.j. ak $\emptyset \neq A\subseteq \mathbf{R}^{n}$ budeme sa zaobera\v{t} funkciami $f:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R} $. Preto\v{z}e prvkami oboru hodn\^{o}t $f(A)$ s\'{u} body z $\mathbf{R,}$ kde je mo\v{z}n\'{e} zavies\v{t} obvykl\'{e} usporiadanie, preto m\'{a} zmysel hovori\v{t} o najv\"{a}\v{c}\v{s}ej aj najmen\v{s}ej hodnote. Samozrejme, \v{z}e nemus\'{\i} existova\v{t} ani najv\"{a}\v{c}\v{s}ia ani najmen\v{s}ia hodnota. \begin{definition} Funkcia $f:G\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}$ nadob\'{u}da na mno\v{z}ine $A\subset G$ \emph{maximum (minimum)} v bode $% \mathbf{a}\in A,$ak \[ f(\mathbf{x})\leq f(\mathbf{a})\;\left( f(\mathbf{x})\geq f(\mathbf{a}% )\right) ,\,\forall \mathbf{x}\in A, \]% \v{c}o zapisujeme% \[ \max_{\mathbf{x}\in A}f(\mathbf{x})=f(\mathbf{a}),\,(\min_{\mathbf{x}\in A}f(% \mathbf{x})=f(\mathbf{a})). \] \end{definition} \begin{definition} Hovor\'{\i}me, \v{z}e $f:G\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}$ m\'{a} \emph{lok\'{a}lne maximum (minimum)} v bode $\mathbf{a}\in G,$ak existuje $O_{\delta }^{{}}\left( \mathbf{a}\right) \subset G$ tak\'{e}, \v{z}e \[ f(\mathbf{x})\leq f(\mathbf{a})\;\left( f(\mathbf{x})\geq f(\mathbf{a}% )\right) ,\,\forall \mathbf{x}\in O_{\delta }\left( \mathbf{a}\right) , \]% \v{c}o zapisujeme \[ \max_{\mathbf{x}\in O_{\delta }\left( \mathbf{a}\right) }f(\mathbf{x})=f(% \mathbf{a}),\,(\min_{\mathbf{x}\in O_{\delta }\left( \mathbf{a}\right) }f(% \mathbf{x})=f(\mathbf{a})). \]% Maxim\'{a} a minim\'{a}, lok\'{a}lne maxim\'{a} a minim\'{a} naz\'{y}vame \emph{extr\'{e}my funkcie }$f.$\emph{\ } \end{definition} Pripome\v{n}me %TCIMACRO{\hyperref{vetu}{}{}{Ma23.tex#3}}% %BeginExpansion \msihyperref{vetu}{}{}{Ma23.tex#3}% %EndExpansion : ak $K\subset \mathbf{R}^{n}$ je kompaktn\'{a} a $f:K\longrightarrow \mathbf{R}$ je spojit\'{a}, potom $f$ nadob\'{u}da na $K$ maximum aj minimum. T\'{a}to veta v\v{s}ak ned\'{a}va postup ako n\'{a}js\v{t} extr\'{e}% my. \begin{theorem} \label{4}Nech $f:G\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}$ je diferencovate\v{l}n\'{a} na otvorenej mno\v{z}ine $G$ a nech $% A\subset G.$ Ak $f$ nadob\'{u}da maximum (minmum) na $A$ vo vn\'{u}tornom bode $\mathbf{a}\in int\left( A\right) ,$ potom $Df_{\mathbf{a}}=\nabla f(% \mathbf{a})=\mathbf{0}.$ \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO411.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO411.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{1} \begin{definition} Nech $f:G\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}$ je diferencovate\v{l}n\'{a} v bode $\mathbf{a}.$\ Bod $\mathbf{a}$ sa naz\'{y}% va \label{5}\emph{stacion\'{a}rny bod} funkcie $f$ ak $\nabla f(\mathbf{a})=% \mathbf{0}.$ \end{definition} \begin{theorem} Nech $f:G\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}$ je diferencovate\v{l}n\'{a} na otvorenej mno\v{z}ine $G$ a nech $A\subset G.$Ak $f$ nadob\'{u}da maximum (alebo minimum) na $A$ v bode $\mathbf{a}\in A,$ potom bu\v{d} $\mathbf{a}$ je stacion\'{a}rny bod, alebo $\mathbf{a}\in \partial A.$ \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO412.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO412.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{2} N\'{a}jdenie stacion\'{a}rnych bodov je rutinn\'{a} z\'{a}le\v{z}itos\v{t}. Vypo\v{c}\'{\i}tame v\v{s}etky parci\'{a}lne deriv\'{a}cie funkcie $f$ a polo% \v{z}\'{\i}me ich (naraz) rovn\'{e} nule. Potom je to \v{c}isto algebrick% \'{a} \'{u}loha. Av\v{s}ak nie v\v{s}etky stacion\'{a}rne body s\'{u} nutne extr\'{e}mami. \begin{example} Funkcia $f:\left\langle -1,1\right\rangle \longrightarrow \mathbf{R}% ,\,f\left( x\right) =x^{3}$ m\'{a} v bode $x=0$ stacion\'{a}rny bod, ale nie je to extr\'{e}m funkcie $f.$ \end{example} \begin{solution} M\'{a}me: $f\,^{\prime }(0)=0,$ teda bod $x=0$ je stacion\'{a}rny bod funkcie $f,$\ ale v tomto bode nie je extr\'{e}m funkcie $f.$ \end{solution} Pre funkciu dvoch premenn\'{y}ch jej graf $z=f\left( x,y\right) $ je plochou v $\mathbf{R}^{3},$ ktor\'{u} si m\^{o}\v{z}eme predstavova\v{t} ako ter\'{e}% n. Maxim\'{a} s\'{u} vrcholky h\^{o}r, minim\'{a} s\'{u} dn\'{a} \'{u}dol% \'{\i}. M\^{o}\v{z}me tam n\'{a}js\v{t} aj horsk\'{e} priesmyky - body, v ktor\'{y}ch je gradient nulov\'{y}, ale vzh\v{l}adom k okolit\'{y}m vrcholom s\'{u} bodmi minima, vzh\v{l}adom k okolit\'{y}m \'{u}doliam s\'{u} zas bodmi maxima. Tak\'{e} body so svojim okol\'{\i}m pripom\'{\i}naj\'{u} konsk% \'{e} sedlo. Ak uva\v{z}ujeme funkciu viac ako \ dvoch premenn\'{y}ch je mo% \v{z}n\'{e} aj in\'{e} chovanie tak\'{y}chto bodov. Stacion\'{a}rne body funkcie, ktor\'{e} nie s\'{u} ani jej (lok\'{a}lnymi)\ maximami ani minimami budeme naz\'{y}va\v{t} \label{6}\emph{sedlov\'{e} body.} \begin{theorem} Lok\'{a}lne extr\'{e}my diferencovate\v{l}nej funkcie sa v\v{z}dy nach\'{a}% dzaj\'{u} v stacion\'{a}rnych bodoch. \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO413.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO413.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{3} Opa\v{c}n\'{e} tvrdenie k predch\'{a}dzaj\'{u}cej vete je nepravdiv\'{e}. Okrem toho nemus\'{\i} existova\v{t} \v{z}iadny lok\'{a}lny extr\'{e}m na uzavretej ohrani\v{c}enej mno\v{z}ine $A,$ aj ke\v{d} glob\'{a}lne extr\'{e}% my existuj\'{u}. \begin{example} Nech $f:A\longrightarrow \mathbf{R},f\left( x,y\right) =x+y,$ kde $% A=\{(x,y)\in \mathbf{R}^{2};\,x^{2}+y^{2}\leq 1\}.$ Funkcia $f$ nem\'{a} na mno\v{z}ine $A$ lok\'{a}lny extr\'{e}m, ale m\'{a} minimum aj maximum. \end{example} \begin{solution} \v{L}ahko sa mo\v{z}no presved\v{c}i\v{t} o tom, \v{z}e \[ f\left( \frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right) =\sqrt{2}=\max_{\left( x,y\right) \in A}f\left( x,y\right) \]% a% \[ f\left( -\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right) =-\sqrt{2}% =\min_{\left( x,y\right) \in A}f\left( x,y\right) . \]% Oba body $\left( \frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right) ,\,\left( -% \frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right) \in \partial A.$\ Funkcia nem% \'{a} lok\'{a}lne extr\'{e}my.\FRAME{dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{% \special{language "Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-3.1416";xmax "3.1416";ymin "0";ymax "1.1";xviewmin "-1.10219999997029";xviewmax "1.10219999999997";yviewmin "-1.10219999999256";yviewmax "1.10219999999256";zviewmin "-1.55874618842399";zviewmax "1.55874618844498";phi 45;theta -21;plottype 14;num-x-gridlines 25;num-y-gridlines 25;plotstyle "patch";axesstyle "normal";plotshading "ZHUE";xis \TEXUX{v58130};yis \TEXUX{z};zis \TEXUX{r};var1name \TEXUX{$\theta $};var2name \TEXUX{$z$};var3name \TEXUX{$r$};function \TEXUX{$\left( r,\theta ,r\cos \theta +r\sin \theta \right) $};linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";coordinateSystem "cylindrical";var1range "-3.1416,3.1416";var2range "0,1.1";surfaceColor "[rainbow:Z:RGB:0x00ff0000:0x000000ff]";surfaceStyle "Color Patch";num-x-gridlines 25;num-y-gridlines 25;surfaceMesh "Mesh";rangeset"Y";function \TEXUX{$\left( r,\theta ,0\right) $};linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";coordinateSystem "cylindrical";var1range "-3.1416,3.1416";var2range "0,1.1";surfaceColor "[rainbow:Z:RGB:0x00ff0000:0x000000ff]";surfaceStyle "Color Patch";num-x-gridlines 25;num-y-gridlines 25;surfaceMesh "Mesh";valid_file "T";tempfilename 'HTRDVV00.wmf';tempfile-properties "XPR";}} $\square $ \end{solution} H\v{l}adanie v\v{s}etk\'{y}ch stacion\'{a}rnych bodov n\'{a}m d\'{a} \'{u}pln% \'{y} zoznam v\v{s}etk\'{y}ch kandid\'{a}tov na lok\'{a}lne extr\'{e}my diferencovate\v{l}nej funkcie. \begin{example} Nech $f:\mathbf{R}^{2}\longrightarrow \mathbf{R},\,f\left( x,y\right) =x^{2}-y^{2}.$ Potom $(0,0)$ je jedin\'{y} stacion\'{a}rny bod, ale nie je to ani minimum ani maximum. \end{example} \begin{solution} M\'{a}me $\frac{\partial f}{\partial x}=2x,\,\frac{\partial f}{\partial y}% =-2y,$ t.j. $\nabla f\left( x,y\right) =\mathbf{0}\Longrightarrow x=0,\,y=0,$ teda $(0,0)$ je stacion\'{a}rny bod. Plat\'{\i}: $f(0,0)=0,$ ale $% f(x,0)=x^{2}>0,$ zatia\v{l} \v{c}o $f(0,y)=-y^{2}.$ Tak bod $(0,0)$ je sedlov% \'{y} bod. Skuto\v{c}ne plocha $z=x^{2}-y^{2}$ najlep\v{s}ie pripom\'{\i}na sedlo.\FRAME{dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";ymin "-5";ymax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-5.01";yviewmax "5.01";zviewmin "-25.05";zviewmax "25.05";phi 46;theta 61;plottype 5;num-x-gridlines 25;num-y-gridlines 25;plotstyle "patch";axesstyle "normal";plotshading "ZHUE";xis \TEXUX{x};yis \TEXUX{y};zis \TEXUX{z};var1name \TEXUX{$x$};var2name \TEXUX{$y$};var3name \TEXUX{$z$};function \TEXUX{$x^{2}-y^{2}$};linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-5,5";var2range "-5,5";surfaceColor "[rainbow:Z:RGB:0x00ff0000:0x000000ff]";surfaceStyle "Color Patch";num-x-gridlines 25;num-y-gridlines 25;surfaceMesh "Mesh";valid_file "T";tempfilename 'HTRDXA01.wmf';tempfile-properties "XPR";}% } $\square $ \end{solution} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma4.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma42.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma42.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O4.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C4.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \section{Maximum a minimum re\'{a}lnej funkcie viacer\'{y}ch premenn\'{y}ch} \end{document}