%% 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 16:59:38} %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 - Nutn\U{e1} a posta\U{10d}uj\U{fa}ca podmienka existencie extr\U{e9}mu\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|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{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma42.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma42.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma45.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma45.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} \subsection{Nutn\'{a} a posta\v{c}uj\'{u}ca podmienka existencie extr\'{e}mu.% } V tejto \v{c}asti budeme formulova\v{t} vety, na z\'{a}klade, ktor\'{y}ch dok% \'{a}\v{z}eme ur\v{c}i\v{t} druh extr\'{e}mu re\'{a}lnej funkcie viacer\'{y}% ch premenn\'{y}ch. \begin{theorem} \label{3}(Nutn\'{a} a posta\v{c}uj\'{u}ca podmienka existencie lok\'{a}lneho extr\'{e}mu) Nech $G\subseteq \mathbf{R}^{n}$ je otvoren\'{a} a nech $f\in C^{2}(G).$ Nech $\mathbf{a}\in G$ je stacion\'{a}rny bod funkcie $% f:G\longrightarrow \mathbf{R}$ a nech $D^{2}f_{\mathbf{x}}(\mathbf{h})$ je kvadratick\'{a} forma (druh\'{y} diferenci\'{a}l funkcie $f$ v bode $\mathbf{% x}$) \[ D^{2}f_{\mathbf{x}}(\mathbf{h})=\sum_{i,j=1}^{n}\frac{\partial ^{2}f(\mathbf{% x})}{\partial x_{i}\partial x_{j}}h_{i}h_{j}. \]% Potom ak a) $D^{2}f_{\mathbf{a}}(\mathbf{h})$ je kladne definitn\'{a}, funkcia $f$ m% \'{a} v bode $\mathbf{a}$ ostr\'{e} lok\'{a}lne minimum $\min f(\mathbf{x}% )=f(\mathbf{a}),$ b) $D^{2}f_{\mathbf{a}}(\mathbf{h})$ je z\'{a}porne definitn\'{a}, funkcia $% f $ m\'{a} v bode $\mathbf{a}$ ostr\'{e} lok\'{a}lne maximum $\max f(\mathbf{% x})=f(\mathbf{a}),$ c) $D^{2}f_{\mathbf{a}}(\mathbf{h})$ je indefinitn\'{a}, bod $\mathbf{a}$ je sedlov\'{y} bod. Opa\v{c}ne: a') Ak je v bode $\mathbf{a}$ ostr\'{e} lok\'{a}lne minimum funkcie $f$, tak kvadratick\'{a} forma $D^{2}f_{\mathbf{a}}(\mathbf{h})$ je kladne semidefinitn\'{a}, b') Ak je v bode $\mathbf{a}$ je ostr\'{e} lok\'{a}lne maximum funkcie $f$, tak $D^{2}f_{\mathbf{a}}(\mathbf{h})$ je z\'{a}porne semidefinitn\'{a}. \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO431.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO431.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{1} \v{C}asti a) - c) vety s\'{u} posta\v{c}uj\'{u}ce podmienky pre to, aby stacion\'{a}rny bod bol minimom, maximom alebo sedlov\'{y}m bodom, zatia\v{l} \v{c}o a'), b') s\'{u} nutn\'{e} podmienky. Predch\'{a}dzaj\'{u}ca veta hovor% \'{\i} o tom, \v{z}e stacion\'{a}rny bod je extr\'{e}mom funkcie v pr\'{\i}% pade, \v{z}e determinant hessi\'{a}nu je nenulov\'{y}. Ak $detH=0$ v bode $% \mathbf{a,}$ potom hovor\'{\i}me o \label{4}\emph{degenerovanom stacion\'{a}% rnom bode} a vtedy ur\v{c}enie druhu extr\'{e}mu z\'{a}vis\'{\i} od diferenci% \'{a}lov vy\v{s}\v{s}\'{\i}ch r\'{a}dov (my sa v\v{s}ak s tak\'{y}m pr\'{\i}% padom v na\v{s}om kurze nebudeme zaobera\v{t}). V nasleduj\'{u}com pr\'{\i}% klade uk\'{a}\v{z}eme ako sa d\'{a} ur\v{c}i\v{t} druh extr\'{e}mu aj v pr% \'{\i}pade degenerovan\'{e}ho stacion\'{a}rneho bodu. \begin{example} N\'{a}jdite stacion\'{a}rne body a ur\v{c}te ich druh pre funkciu \[ f:\mathbf{R}^{2}\longrightarrow \mathbf{R},\,f\left( x,y\right) =x^{4}+y^{4}+6x^{2}y^{2}+8x^{3}. \] \end{example} \begin{solution} M\'{a}me \[ \frac{\partial f}{\partial x}=4x^{3}+12xy^{2}+24x^{2},\,\frac{\partial f}{% \partial y}=4y^{3}+12x^{2}y=4y(y^{2}+3x^{2}). \]% Stacion\'{a}rne body s\'{u} $(0,0),\,(-6,0).$ Hessi\'{a}n bude \[ H=\left( \begin{array}{cc} \frac{\partial ^{2}f}{\partial x^{2}} & \frac{\partial ^{2}f}{\partial y\partial x} \\ \frac{\partial ^{2}f}{\partial x\partial y} & \frac{\partial ^{2}f}{\partial y^{2}}% \end{array}% \right) =\left( \begin{array}{cc} 12x^{2}+12y^{2}+48x & 24xy \\ 24xy & 12y^{2}+12x^{2}% \end{array}% \right) . \]% V bode $(-6,0)$ je \[ H(-6,0)=\left( \begin{array}{cc} 144 & 0 \\ 0 & 32% \end{array}% \right) \]% a kvadratick\'{a} forma je \[ Q(\mathbf{a},\mathbf{h})=Q((-6,0),(h,k))=(h,k)\left( \begin{array}{cc} 144 & 0 \\ 0 & 32% \end{array}% \right) \left( \begin{array}{c} h \\ k% \end{array}% \right) =144h^{2}+432k^{k}>0 \]% t.j. v bode $(-6,0)$ je lok\'{a}lne minimum $\min f(\mathbf{x}% )=f(-6,0)=2(-6)^{3}.$ V bode $(0,0)$ je \[ H(0,0)=\left( \begin{array}{cc} 0 & 0 \\ 0 & 0% \end{array}% \right) \]% teda $(0,0)$ je degenerovan\'{y} stacion\'{a}rny bod a ur\v{c}enie lok\'{a}% lneho extr\'{e}mu pomocou druh\'{e}ho diferenci\'{a}lu nie je mo\v{z}n\'{e}. V tomto pr\'{\i}pade v\v{s}ak mo\v{z}no \v{l}ahko overi\v{t}, \v{z}e \[ f(h,k)-f(0,0)=h^{4}+k^{4}+6h^{2}k^{2}+8h^{3}=(h^{2}+k^{2})^{2}+4h^{2}k^{2}+8h^{3} \]% odkia\v{l} vidie\v{t}, \v{z}e ak $h=0$ a $k$ je \v{l}ubovo\v{l}n\'{e}, tak rozdiel je kladn\'{y} a pre $(-h,0)$ je rozdiel z\'{a}porn\'{y}, t.j. $(0,0)$ je sedlov\'{y} bod (funkcia nem\'{a} extr\'{e}m). $\square $ \end{solution} Ak m\'{a}me funkciu dvoch premenn\'{y}ch $f\left( x,y\right) $ potom hessi% \'{a}n v bode $\mathbf{a}$ \ a druh\'{y} diferenci\'{a}l v bode $\mathbf{a}$ v smere $\mathbf{h=}\left( h,k\right) $ s\'{u} definovan\'{e}: \[ H\left( \mathbf{a}\right) =\left( \begin{array}{cc} \frac{\partial ^{2}f\left( \mathbf{a}\right) }{\partial x^{2}} & \frac{% \partial ^{2}f\left( \mathbf{a}\right) }{\partial y\partial x} \\ \frac{\partial ^{2}f\left( \mathbf{a}\right) }{\partial x\partial y} & \frac{% \partial ^{2}f\left( \mathbf{a}\right) }{\partial y^{2}}% \end{array}% \right) ,\,D^{2}f_{\mathbf{a}}\left( \mathbf{h}\right) =\frac{\partial ^{2}f\left( \mathbf{a}\right) }{\partial x^{2}}h_{{}}^{2}+2\frac{\partial ^{2}f\left( \mathbf{a}\right) }{\partial y\partial x}hk+\frac{\partial ^{2}f\left( \mathbf{a}\right) }{\partial y^{2}}k_{{}}^{2} \]% a my dost\'{a}vame tvrdenia ekvivalentn\'{e} s predch\'{a}dzaj\'{u}cou vetou. \begin{theorem} Nech $\det H\left( \mathbf{a}\right) \neq 0.$ Druh\'{y} diferenci\'{a}l funkcie $f:G\left( \subset \mathbf{R}^{2}\right) \longrightarrow \mathbf{R}$ \ v stacion\'{a}rnom bode $\mathbf{a}$\ v smere $\mathbf{h=}\left( h,k\right) $% \[ \,D^{2}f_{\mathbf{a}}\left( \mathbf{h}\right) =\frac{\partial ^{2}f\left( \mathbf{a}\right) }{\partial x^{2}}h_{{}}^{2}+2\frac{\partial ^{2}f\left( \mathbf{a}\right) }{\partial y\partial x}hk+\frac{\partial ^{2}f\left( \mathbf{a}\right) }{\partial y^{2}}k_{{}}^{2} \] je a) kladne definitn\'{y}, vtedy a len vtedy ak \[ \frac{\partial ^{2}f\left( \mathbf{a}\right) }{\partial x^{2}}>0\text{ \ a \ }\det H\left( \mathbf{a}\right) =\frac{\partial ^{2}f\left( \mathbf{a}% \right) }{\partial x^{2}}\frac{\partial ^{2}f\left( \mathbf{a}\right) }{% \partial y^{2}}-\left( \frac{\partial ^{2}f\left( \mathbf{a}\right) }{% \partial y\partial x}\right) ^{2}>0, \]% potom v stacion\'{a}rnom bode $\mathbf{a}$\ funkcie $f$ je ostr\'{e} lok\'{a}% lne minimum funkcie $f,$ b) z\'{a}porne definitn\'{y}, vtedy a len vtedy ak \[ \frac{\partial ^{2}f\left( \mathbf{a}\right) }{\partial x^{2}}<0\text{ \ a \ }\det H\left( \mathbf{a}\right) =\frac{\partial ^{2}f\left( \mathbf{a}% \right) }{\partial x^{2}}\frac{\partial ^{2}f\left( \mathbf{a}\right) }{% \partial y^{2}}-\left( \frac{\partial ^{2}f\left( \mathbf{a}\right) }{% \partial y\partial x}\right) ^{2}>0, \]% potom v stacion\'{a}rnom bode $\mathbf{a}$\ funkcie $f$ je ostr\'{e} lok\'{a}% lne maximum funkcie $f,$ c) indefinitn\'{y}, vtedy a len vtedy ak \[ \det H\left( \mathbf{a}\right) =\frac{\partial ^{2}f\left( \mathbf{a}\right) }{\partial x^{2}}\frac{\partial ^{2}f\left( \mathbf{a}\right) }{\partial y^{2}}-\left( \frac{\partial ^{2}f\left( \mathbf{a}\right) }{\partial y\partial x}\right) ^{2}<0, \]% potom je stacion\'{a}rny bod $\mathbf{a}$\ funkcie $f$ sedlov\'{y} bod. \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO432.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO432.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{2} \begin{example} N\'{a}jdite stacion\'{a}rne body funkcie $f:\mathbf{R}^{2}\longrightarrow \mathbf{R},\,f\left( x,y\right) =x^{2}y+xy^{2}-3xy$ a ur\v{c}te ich druh. \end{example} \begin{solution} $\frac{\partial f}{\partial x}\left( x,y\right) =2xy+y^{2}-3y,\frac{\partial f}{\partial y}\left( x,y\right) =x^{2}+2xy-3x.$ Ak polo\v{z}\'{\i}me \[ \nabla f=\left( \frac{\partial f}{\partial x},\frac{\partial f}{\partial y}% \right) =(0,0), \]% dostaneme stacion\'{a}rne body $(0,0),\,(1,1),\,(3,0),\,(0,3).$ Potom m\'{a}% me \[ \frac{\partial ^{2}f}{\partial x^{2}}=2y,\,\frac{\partial ^{2}f}{\partial y\partial x}=2x+2y-3,\,\frac{\partial ^{2}f}{\partial y^{2}}=2x. \]% Tak pre bod: $(0,0)$ je $\frac{\partial ^{2}f}{\partial x^{2}}(0,0)=0$ a $\det H(0,0)=-9<0 $ sedlo, $(1,1)$ je $\frac{\partial ^{2}f}{\partial x^{2}}(1,1)=2$ a $\det H(1,1)=3>0$ lok\'{a}lne minimum $\min f(\mathbf{x})=f(1,1)=-1,$ $(3,0)$ je $\frac{\partial ^{2}f}{\partial x^{2}}(3,0)=0$ a $\det H(3,0)=-9<0 $ sedlo, $(0,3)$ je $\frac{\partial ^{2}f}{\partial x^{2}}(0,3)=0$ a $\det H(0,3)=-9<0 $ sedlo. Pou\v{z}it\'{\i}m druh\'{e}ho diferenci\'{a}lu (kvadratickej formy) dostaneme samozrejme ten ist\'{y} v\'{y}sledok, ale pre lep\v{s}\'{\i} preh% \v{l}ad ho tu uvedieme. V bode $(0,0)$ je kvadratick\'{a} forma $D^{2}f_{\left( 0,0\right) }\left( \left( h,k\right) \right) =-6hk$ indefinitn\'{a}, $(1,1)$ je $D^{2}f_{\left( 1,1\right) }\left( \left( h,k\right) \right) =2h^{2}+2hk+2k^{2}=2\left( h+\frac{k}{2}\right) ^{2}+\frac{3}{2}k^{2}$ kladne definitn\'{a}, lok\'{a}lne minimum $\min f(\mathbf{x})=f(1,1)=-1,$ $(3,0)$ je $D^{2}f_{\left( 3,0\right) }\left( \left( h,k\right) \right) =6% \left[ \left( k+\frac{h}{2}\right) ^{2}-\frac{1}{4}h^{2}\right] $ indefinitn% \'{a}, $(0,3)$ je$\ D^{2}f_{\left( 0,3\right) }\left( \left( h,k\right) \right) =% \left[ \left( h+\frac{k}{2}\right) ^{2}-\frac{1}{4}k^{2}\right] $ indefinitn% \'{a}. $\square $ \end{solution} \v{D}al\v{s}ou met\'{o}dou ako ur\v{c}i\v{t}, \v{c}i je druh\'{y} diferenci% \'{a}l kladne, alebo z\'{a}porne definitn\'{y} je \emph{Sylvestrovo krit\'{e}% rium,} ktor\'{e} nebudeme dokazova\v{t}, jeho d\^{o}kaz plynie z v\'{y}% sledkov line\'{a}rnej algebry. \begin{theorem} \label{5}(Sylvestrovo krit\'{e}rium) Pre funkciu $f:G\left( \subset \mathbf{R% }^{n}\right) \longrightarrow \mathbf{R},\,f\in C^{2}(G)$ druh\'{y} diferenci% \'{a}l $D^{2}f_{\mathbf{a}}\left( \mathbf{h}\right) ,\mathbf{h\neq 0}$ je kladne (z\'{a}porne) definitn\'{a} kvadratick\'{a} forma vtedy a len vtedy ak \[ \Delta _{k}>0,\left( (-1)^{k}\Delta _{k}>0\right) ,k=1,2,\dots ,n \]% kde \[ \Delta _{k}=\left| \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1k} \\ a_{21} & a_{22} & \dots & a_{2k} \\ \vdots & \vdots & \ddots & \vdots \\ a_{k1} & a_{k2} & \dots & a_{kk}% \end{array}% \right| ,\text{ \ pri\v{c}om \ }a_{ij}=\frac{\partial ^{2}f(\mathbf{a})}{% \partial x_{j}\partial x_{i}}=a_{ji}. \]% Ak $\Delta _{n}\neq 0$ a kvadratick\'{a} forma nie je definitn\'{a}, potom je indefinitn\'{a}. \end{theorem} \begin{example} N\'{a}jdite extr\'{e}my funkcie \[ f:\mathbf{R}^{3}\longrightarrow \mathbf{R}% ,\,f(x,y,z)=x^{3}+y^{2}+z^{2}+12xy+2z. \] \end{example} \begin{solution} \[ \frac{\partial f}{\partial x}(x,y,z)=3x^{2}+12y, \]% \[ \frac{\partial f}{\partial y}(x,y,z)=2y+12x, \]% \[ \frac{\partial f}{\partial z}(x,y,z)=2z+2. \]% V kritickom bode plat\'{\i} \[ \nabla f(x,y,z)=\mathbf{0}\text{ \ t.j. \ \ }\frac{\partial f}{\partial x}% (x,y,z)=\frac{\partial f}{\partial y}(x,y,z)=\frac{\partial f}{\partial z}% (x,y,z)=0. \]% Potom \[ z=-1,\,y=-6x,\,3x^{2}-72x=0\Longrightarrow x_{1}=0,x_{2}=24. \]% Stacion\'{a}rne body s\'{u} $\mathbf{a}^{1}=(0,0,-1),\,\mathbf{a}% ^{2}=(24,-144,-1),$ \v{d}alej m\'{a}me \[ \frac{\partial ^{2}f}{\partial x^{2}}(x,y,z)=6x,\,\frac{\partial ^{2}f}{% \partial x\partial y}(x,y,z)=12,\,\frac{\partial ^{2}f}{\partial x\partial z}% (x,y,z)=0, \]% \[ \frac{\partial ^{2}f}{\partial y^{2}}(x,y,z)=2,\,\frac{\partial ^{2}f}{% \partial y\partial z}(x,y,z)=0,\,\frac{\partial ^{2}f}{\partial z^{2}}% (x,y,z)=2. \]% Potom v bode $\mathbf{a}^{1}=(0,0,-1)$ pod\v{l}a Sylvestrovho krit\'{e}ria \[ \Delta _{3}=\left| \begin{array}{ccc} 0 & 12 & 0 \\ 12 & 2 & 0 \\ 0 & 0 & 2% \end{array}% \right| =-288<0,\,\Delta _{2}=\left| \begin{array}{cc} 0 & 12 \\ 12 & 2% \end{array}% \right| =-144<0,\,\Delta _{1}=\left| \begin{array}{c} 0% \end{array}% \right| =0, \]% nie je extr\'{e}m, je to sedlov\'{y} bod. To ist\'{e} tvrdenie dostaneme, ak pou\v{z}ijeme druh\'{y} diferenci\'{a}l: \[ D^{2}f_{\mathbf{a}^{1}}(\mathbf{h})=\sum_{i,j=1}^{3}\frac{\partial ^{2}f(% \mathbf{a}^{1})}{\partial x_{i}\partial x_{j}}% h_{i}h_{j}=24h_{1}h_{2}+2h_{2}^{2}+2h_{3}^{2}, \]% \ odkia\v{l} plynie, \v{z}e pre $\mathbf{h}^{1}=\left( 1,-1,1\right) $\ m% \'{a}me: $D^{2}f_{\mathbf{a}^{1}}(\mathbf{h})=-20,\,$pre $\mathbf{h}% ^{2}=\left( 1,1,1\right) $\ m\'{a}me: $D^{2}f_{\mathbf{a}^{1}}(\mathbf{h}% )=28.$ Druh\'{y} diferenci\'{a}l je indefinitn\'{a} kvadratick\'{a} forma a bod $\mathbf{a}^{1}$je sedlov\'{y} bod. V bode $\mathbf{a}^{2}=(24,-144,-1)$ pod\v{l}a Sylvestrovho krit\'{e}ria \[ \Delta _{3}=\left| \begin{array}{ccc} 144 & 12 & 0 \\ 12 & 2 & 0 \\ 0 & 0 & 2% \end{array}% \right| =288>0,\,\Delta _{2}=\left| \begin{array}{cc} 144 & 12 \\ 12 & 2% \end{array}% \right| =144>0,\,\Delta _{1}=\left| 144\right| =144>0, \]% je $f\left( 24,-144,-1\right) =\allowbreak -6913$ lok\'{a}lne (relat\'{\i}% vne) minimum. Ak pou\v{z}ijeme druh\'{y} diferenci\'{a}l: \[ D^{2}f_{\mathbf{a}^{2}}(\mathbf{h})=\sum_{i,j=1}^{3}\frac{\partial ^{2}f(% \mathbf{a}^{2})}{\partial x_{i}\partial x_{j}}% h_{i}h_{j}=144h_{1}^{2}+24h_{1}h_{2}+2h_{2}^{2}+2h_{3}^{2}=2\left[ \left( 6h_{1}+h_{2}\right) ^{2}+36h_{1}^{2}+h_{3}^{2}\right] >0,\,\forall \mathbf{h}% \neq \mathbf{0}, \]% \ odkia\v{l} plynie, \v{z}e druh\'{y} diferenci\'{a}l je kladne definitn\'{a} kvadratick\'{a} forma a v bode $\mathbf{a}^{2}$ funkcia nadob\'{u}da lok\'{a}% lne minimum $f\left( 24,-144,-1\right) =\allowbreak -6913.$ $\square $ \end{solution} Vhodn\'{y}m pr\'{\i}kladom je aj met\'{o}da najmen\v{s}\'{\i}ch \v{s}% tvorcov, t.j. \'{u}loha prelo\v{z}i\v{t} priamku $y=mx+c$ bodmi $% (x_{1},y_{1}),(x_{2},y_{2}),\dots ,(x_{p},y_{p}).$ Pritom chceme aby $% f(m,c)=\sum_{i=1}^{p}(y_{i}-mx_{i}-c)^{2}$ bola \v{c}o najmen\v{s}ia. \begin{description} \item[Pozn\'{a}mka] Ak test druhou deriv\'{a}ciou zlyh\'{a} v bode $\mathbf{a% }=(a,b,c),$ potom treba sk\'{u}ma\v{t} priamo rozdiel $f\left( \mathbf{a}+% \mathbf{h}\right) -f\left( \mathbf{a}\right) =f(a+h,b+k,c+l)-f(a,b,c)$ pre ka% \v{z}d\'{e} $\mathbf{h}=\left( h,k,l\right) .$ \end{description} \begin{example} Uk\'{a}\v{z}te, \v{z}e jedin\'{y} stacion\'{a}rny bod funkcie \[ f:\mathbf{R}^{3}\longrightarrow \mathbf{R},\,f(x,y,z)=\limfunc{arctg}% (x^{2}+y^{2}+z^{2})-(xy+yz+xz) \] je bod $(0,0,0)$ a ur\v{c}te jeho druh. \end{example} \begin{solution} \[ \frac{\partial f}{\partial x}(x,y,z)=\frac{2x}{1+(x^{2}+y^{2}+z^{2})^{2}}% -(y+z), \]% \[ \frac{\partial f}{\partial y}(x,y,z)=\frac{2y}{1+(x^{2}+y^{2}+z^{2})^{2}}% -(z+x), \]% \[ \frac{\partial f}{\partial z}(x,y,z)=\frac{2z}{1+(x^{2}+y^{2}+z^{2})^{2}}% -(x+y). \]% V kritickom bode plat\'{\i} \[ \nabla f(x,y,z)=\mathbf{0}\text{ \ t.j. \ \ }\frac{\partial f}{\partial x}% (x,y,z)=\frac{\partial f}{\partial y}(x,y,z)=\frac{\partial f}{\partial z}% (x,y,z)=0. \]% Potom aj \[ 0=\frac{\partial f}{\partial x}(x,y,z)-\frac{\partial f}{\partial y}% (x,y,z)=(x-y)\left[ 1+\frac{2}{1+(x^{2}+y^{2}+z^{2})^{2}}\right] ,\,\text{% t.j. \ }x=y. \]% Preto\v{z}e vz\v{t}ahy s\'{u} symetrick\'{e}, dostaneme napokon $x=y=z.$ Teda stacion\'{a}rny bod bude ma\v{t} s\'{u}radnice $(x,x,x),$ odkia\v{l} \[ \frac{2x}{1+9x^{4}}-2x=0,\,\text{t.j. }x=0. \]% Teda $(0,0,0)$ je stacion\'{a}rny bod. Ak urob\'{\i}me test druhou deriv\'{a}% ciou, tento zlyh\'{a} (presved\v{c}te sa o tom). Vyjadr\'{\i}me si rozdiel \[ f(h,k,l)-f(0,0,0)=\limfunc{arctg}\left( h^{2}+k^{2}+l^{2}\right) -\left( hk+kl+hl\right) \]% Pre funkciu $\limfunc{arctg}u$ pou\v{z}ijeme Taylorov rozvoj: \[ \limfunc{arctg}u=u-\frac{u^{3}}{3}+\frac{u^{5}}{5}-\dots ,\,\text{\ \ pre \ }% \left| u\right| <1. \]% Potom pre $(h,k,l)$ bl\'{\i}zko $(0,0,0)$ m\'{a}me \[ f(h,k,l)-f(0,0,0)=h^{2}+k^{2}+l^{2}-\frac{(h^{2}+k^{2}+l^{2})^{3}}{3}% -(hk+kl+hl)+E, \]% kde $E$ pozost\'{a}va z v\'{y}razov vy\v{s}\v{s}\'{\i}ch r\'{a}dov. Potom pre dostato\v{c}ne mal\'{e} $h$ m\'{a}me \[ f(h,0,0)=h^{2}-\frac{h^{6}}{3}>0,\,\ \ \text{zatia\v{l} \v{c}o \ }% f(h,h,h)=-9h^{6}<0. \]% Teda bod $(0,0,0)$ je sedlov\'{y} bod. $\square $ \end{solution} \begin{center} \begin{tabular}{|c|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{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma42.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma42.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma45.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma45.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}