%% 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 16:28:46} %TCIDATA{} %TCIDATA{} %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 I online - Zbierka 3\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{Zbierka \'{u}loh} \begin{center} \begin{tabular}{|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Zbierka}{}{}{Z.tex}}% %BeginExpansion \msihyperref{Zbierka}{}{}{Z.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \subsection{Priebeh funkcie.} \textbf{Pr\'{\i}klad 1. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\frac{x^{3}}{x^{2}+4}.% \CustomNote{Prob_Solv_Hint}{% {}$D\left( f\right) =\mathbf{R},$ spojit\'{a}, $f\left( 0\right) =0,$ nem% \'{a} asymptoty bez smernice \par Funkcia $f$ je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \frac{x^{2}\left( x^{2}+12\right) }{\left( x^{2}+4\right) ^{2}},$ pre $x\in \left( -\infty ,\infty \right) ,\,f\nearrow ,$ nem\'{a} extr\'{e}my \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{8x\left( 12-x^{2}\right) }{\left( x^{2}+4\right) ^{3}},$ pre $x\in \left( -\infty ,-% \sqrt{12}\right) $ a $\left( 0,\sqrt{12}\right) ,\,f\cup ,$ pre $x\in \left( -\sqrt{12},0\right) $ a $\left( \sqrt{12},\infty \right) ,\,f\cap ,$ v $% x=0,\,\pm \sqrt{12}$ m\'{a} $f$ inflexn\'{e} body, $f\left( 0\right) =0,\,f\left( -\sqrt{12}\right) =-\frac{3}{2}\sqrt{3},\,f\left( \sqrt{12}% \right) =\frac{3}{2}\sqrt{3}\allowbreak .$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =-\infty ,\,\lim_{x\longrightarrow \infty }f\left( x\right) =\infty .$ Asymptota v $% -\infty :y=x,$ asymptota v $\infty :y=x.\,H\left( f\right) =\left( -\infty ,\infty \right) .$% \par Graf \ $f\left( x\right) =\frac{x^{3}}{x^{2}+4}$\FRAME{dtbpFX}{3.2811in}{% 2.1871in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2811in;height 2.1871in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-5.01";yviewmax "5.01";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\frac{x^{3}}{x^{2}+4}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";function \TEXUX{$x$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 2;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU2OJ314.wmf';tempfile-properties "XPR";}% }}$ \textbf{Pr\'{\i}klad 2. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\frac{2x-1}{\left( x-1\right) ^{2}}.% \CustomNote{Prob_Solv_Hint}{% {}$D\left( f\right) =\mathbf{R\setminus }\left\{ 1\right\} ,$ spojit\'{a}, $% f\left( \frac{1}{2}\right) =0,$ asymptota bez smernice $x=1\,\,\left( \lim_{x\longrightarrow 1}f\left( x\right) =\infty \right) .$% \par Funkcia $f$ nie je p\'{a}rna, $f$ nie je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \frac{-2x}{\left( x-1\right) ^{3}},$ pre $x\in \left( 0,1\right) ,\,f\nearrow ,$ pre $x\in \left( -\infty ,0\right) $ a $\left( 1,\infty \right) ,\,f\searrow ,$ $\min f\left( x\right) =f\left( 0\right) =-1.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{2\left( 2x+1\right) }{\left( x^{2}-1\right) ^{4}},$ pre $x\in \left( -\frac{1}{2}% ,1\right) $ a $\left( 1,\infty \right) ,\,f\cup ,$ pre $x\in \left( -\infty ,-\frac{1}{2}\right) ,\,f\cap ,$ v $x=-\frac{1}{2}$ m\'{a} $f$ inflexn\'{y} bod, $f\left( -\frac{1}{2}\right) =-\frac{8}{9}.$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =0,\,\lim_{x\longrightarrow \infty }f\left( x\right) =0.$ Asymptota v $% -\infty :y=0,$ asymptota v $\infty :y=0.\,H\left( f\right) =\left\langle -1,\infty \right) .$% \par Graf \ $f\left( x\right) =\frac{2x-1}{\left( x-1\right) ^{2}}$\FRAME{dtbpFX}{% 3.2707in}{2.1802in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2707in;height 2.1802in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5.01";xmax "5.01";xviewmin "-5.01";xviewmax "5.01";yviewmin "-2";yviewmax "20";viewset"XY";rangeset"X";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\frac{2x-1}{\left( x-1\right) ^{2}}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5.01,5.01";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";function \TEXUX{$0$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-5.01,5.01";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU2OI713.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 3. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =x^{2}e^{\frac{1}{x}}.% \CustomNote{Prob_Solv_Hint}{% {}$D\left( f\right) =\mathbf{R\setminus }\left\{ 0\right\} ,$ spojit\'{a}, nem\'{a} nulov\'{e} body, asymptota bez smernice $x=0$ $\,\,\left( \lim_{x\longrightarrow 0^{-}}f\left( x\right) =0,\,\lim_{x\longrightarrow 0^{+}}f\left( x\right) =\infty \right) .$% \par Funkcia $f$ nie je p\'{a}rna, $f$ nie je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \left( 2x-1\right) e^{\frac{1}{x% }},$ pre $x\in \left( \frac{1}{2},\infty \right) ,\,f\nearrow ,$ pre $x\in \left( -\infty ,0\right) $ a $\left( 0,\frac{1}{2}\right) ,\,f\searrow ,$ $% \limfunc{lokmin}f\left( x\right) =f\left( \frac{1}{2}\right) =\frac{1}{4}% e^{2}.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{2e^{\frac{1}{x}}}{% x^{2}}\left[ \left( x-\frac{1}{2}\right) ^{2}+\frac{1}{4}\right] ,$ pre $% x\in \left( -\infty ,0\right) $ a $\left( 0,\infty \right) ,\,f\cup ,$ nem% \'{a} inflexn\'{y} bod. \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =\infty ,\,\lim_{x\longrightarrow \infty }f\left( x\right) =\infty .$ Nem\'{a} asymptoty v $-\infty ,$ $\infty .\,H\left( f\right) =\left( 0,\infty \right) .$% \par Graf \ $f\left( x\right) =x^{2}e^{\frac{1}{x}}$\FRAME{dtbpFX}{3.2915in}{% 2.194in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2915in;height 2.194in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5.01";xmax "5.01";xviewmin "-5.01";xviewmax "5.01";yviewmin "-0.075502";yviewmax "20";viewset"XY";rangeset"X";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$x^{2}e^{\frac{1}{x}}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5.01,5.01";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";valid_file "T";tempfilename 'HU2OGX12.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 4. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\arcsin \frac{1-x}{1+x}.% \CustomNote{Prob_Solv_Hint}{% {}$D\left( f\right) =\left\langle 0,\infty \right) ,$ spojit\'{a}, $f\left( 1\right) =0,$ nem\'{a} asymptoty bez smernice. \par Funkcia $f$ nie je p\'{a}rna, $f$ nie je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak -\frac{1}{\left( 1+x\right) \sqrt{x}},$ pre $x\in \left( 0,\infty \right) ,\,f\searrow ,$ $\max f\left( x\right) =f\left( 0\right) =\frac{\pi }{2},$ nem\'{a} lok\'{a}lne minimum. \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{3x+1}{2x\sqrt{x}% \left( 1+x\right) ^{2}},$ pre $x\in \left( 0,\infty \right) ,\,f\cup ,$ nem% \'{a} inflexn\'{y} bod. \par $\lim_{x\longrightarrow \infty }f\left( x\right) =-\frac{\pi }{2}.$ Asymptota v $\infty :y=-\frac{\pi }{2}.\,H\left( f\right) =\left( -\frac{\pi }{2},\frac{\pi }{2}\right\rangle .$% \par Graf \ $f\left( x\right) =\arcsin \frac{1-x}{1+x}$\FRAME{dtbpFX}{3.2603in}{% 2.1733in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2603in;height 2.1733in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "0";xmax "10";xviewmin "-0.01";xviewmax "10.01";yviewmin "-1.57393791944849";yviewmax "1.57393791944849";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\arcsin \frac{1-x}{1+x}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "0,10";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";function \TEXUX{$-\frac{\pi }{2}$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0,10";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU2OG911.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 5. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\frac{x^{3}}{2\left( x+1\right) ^{2}}.% \CustomNote{Prob_Solv_Hint}{% {}$D\left( f\right) =\mathbf{R\setminus }\left\{ -1\right\} ,$ spojit\'{a}, $% f\left( 0\right) =0,$ asymptota bez smernice $x=-1\,\,\left( \lim_{x\longrightarrow -1}f\left( x\right) =-\infty \right) .$% \par Funkcia $f$ nie je p\'{a}rna, $f$ nie je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \frac{x^{2}\left( x+3\right) }{% 2\left( x+1\right) ^{3}},$ pre $x\in \left( -\infty ,-3\right) $ a $\left( -1,\infty \right) ,\,f\nearrow ,$ pre $x\in \left( -3,-1\right) ,\,f\searrow ,$ $\limfunc{lokmax}f\left( x\right) =f\left( -3\right) =-\frac{27}{8}.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{3x}{\left( 1+x\right) ^{4}},$ pre $x\in \left( 0,\infty \right) ,\,f\cup ,$ pre $x\in \left( -\infty ,-1\right) $ a $\left( -1,0\right) ,\,f\cap ,$ v $x=0$ m\'{a} $f$ inflexn\'{y} bod., $f\left( 0\right) =0.$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =-\infty ,\,\lim_{x\longrightarrow \infty }f\left( x\right) =\infty .$ Asymptota v $% -\infty :y=\frac{1}{2}x-1,$ asymptota v $\infty :y=\frac{1}{2}x-1.\,H\left( f\right) =\left( -\infty ,\infty \right) .$% \par Graf \ $f\left( x\right) =\frac{x^{3}}{2\left( x+1\right) ^{2}}$\FRAME{dtbpFX% }{3.1566in}{2.1041in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.1566in;height 2.1041in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-51.1378907408562";yviewmax "1.78893229178241";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\frac{x^{3}}{2\left( x+1\right) ^{2}}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";function \TEXUX{$\frac{1}{2}x-1$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU2OFK10.wmf';tempfile-properties "XPR";}% }}$ \textbf{Pr\'{\i}klad 6. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =x^{2}-2\left\vert x\right\vert .% \CustomNote{Prob_Solv_Hint}{% {}$D\left( f\right) =\mathbf{R},$ spojit\'{a}, $f\left( 0\right) =0,$ $% f\left( -2\right) =0,$ $f\left( 2\right) =0$ nem\'{a} asymptoty bez smernice. \par Funkcia $f$ je p\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \left\{ \begin{tabular}{cc} $2x-2$ & $x>0$ \\ $2x+2$ & $x<0$% \end{tabular}% \right. ,\,f\,\,^{\prime }\left( 0\right) $ neexistuje, pre $x\in \left( -1,0\right) $ a $\left( 1,\infty \right) ,\,f\nearrow ,$ pre $x\in \left( -\infty ,-1\right) $ a $\left( 0,1\right) ,\,f\searrow ,$ $\limfunc{lokmax}% f\left( x\right) =f\left( 0\right) =0,\,\limfunc{lokmin}f\left( x\right) =f\left( -1\right) =f\left( 1\right) =-1.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak 2,$ pre $x\in \left( -\infty ,0\right) $ a $\left( 0,\infty \right) ,\,f\cup ,\,f\,\,^{\prime \prime }\left( 0\right) \nexists ,$ nem\'{a} inflexn\'{y} bod. \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =\infty ,\,\lim_{x\longrightarrow \infty }f\left( x\right) =\infty .$ Asymptotu v $% -\infty $ nem\'{a}, asymptotu v $\infty $ nem\'{a}. $\,H\left( f\right) =\left\langle -1,\infty \right) .$% \par Graf \ $f\left( x\right) =x^{2}-2\left\vert x\right\vert .$\FRAME{dtbpFX}{% 3.2396in}{2.1594in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2396in;height 2.1594in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-1.01436588103255";yviewmax "15.0159983675135";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$x^{2}-2\left\vert x\right\vert $};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";valid_file "T";tempfilename 'HU2OES0Z.wmf';tempfile-properties "XPR";}% }}$ \textbf{Pr\'{\i}klad 7. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\frac{x}{x^{2}-4}.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R\setminus }\left\{ -2,2\right\} ,$ spojit\'{a}, $f\left( 0\right) =0,$ asymptota bez smernice $% x=-2,\,x=2\,\,\left( \lim_{x\longrightarrow -2^{-}}f\left( x\right) =-\infty ,\,\lim_{x\longrightarrow -2^{+}}f\left( x\right) =\infty ,\,\lim_{x\longrightarrow 2^{-}}f\left( x\right) =-\infty ,\,\lim_{x\longrightarrow 2^{+}}f\left( x\right) =\infty \right) .$% \par Funkcia $f$ je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak -\frac{4+x^{2}}{\left( x^{2}-4\right) ^{2}},$ pre $x\in \left( -\infty ,-2\right) ,\,\left( -2,2\right) $ a $\left( 2,\infty \right) ,\,f\searrow ,$ nem\'{a} lok\'{a}% lne extr\'{e}my. \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{2x\left( x^{2}+12\right) }{\left( x^{2}-4\right) ^{3}},$ pre $x\in \left( -2,0\right) $ a $\left( 2,\infty \right) ,\,f\cup ,$ pre $x\in \left( -\infty ,-2\right) $ a $\left( 0,2\right) ,\,f\cap ,$ v $x=0$ m\'{a} $f$ inflexn\'{y} bod, $% f\left( 0\right) =0.$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =0,\,\lim_{x\longrightarrow \infty }f\left( x\right) =0.$ Asymptota v $% \infty $ je $y=0,$ \ asymptota v $-\infty $ \ je $y=0.$ $H\left( f\right) =\left( -\infty ,\infty \right) .$ Graf funkcie $f\left( x\right) =\frac{x}{% x^{2}-4}.$\FRAME{dtbpFX}{3.2915in}{2.194in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2915in;height 2.194in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5.01";xmax "5.01";xviewmin "-5.01";xviewmax "5.01";yviewmin "-10";yviewmax "10";viewset"XY";rangeset"X";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\frac{x}{x^{2}-4}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5.01,5.01";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";function \TEXUX{$0$};linecolor "blue";linestyle 1;discont FALSE;pointstyle "point";linethickness 1;lineAttributes "Solid";discont FALSE;var1range "-5.01,5.01";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU2ODQ0Y.wmf';tempfile-properties "XPR";}}}${} \textbf{Pr\'{\i}klad 8. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\frac{10x}{\left( x+2\right) ^{2}}.% \CustomNote{Prob_Solv_Hint}{% {}$D\left( f\right) =\mathbf{R\setminus }\left\{ -2\right\} ,$ spojit\'{a}, $% f\left( 0\right) =0,$ asymptota bez smernice $x=-2\,\,\left( \lim_{x\longrightarrow -2}f\left( x\right) =-\infty \right) .$% \par Funkcia $f$ nie je nep\'{a}rna, funkcia $f$ nie je p\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \frac{10\left( 2-x\right) }{% \left( x+2\right) ^{3}},$ pre $x\in \left( -2,2\right) ,\,f\nearrow ,$ pre $% x\in \left( -\infty ,-2\right) $ a $\left( 2,\infty \right) ,\,f\searrow ,$ $% \max f\left( x\right) =f\left( 2\right) =\frac{5}{4}.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{20\left( x-4\right) }{\left( x+2\right) ^{4}},$ pre $x\in \left( 4,\infty \right) ,\,f\cup ,$ pre $x\in \left( -\infty ,-2\right) $ a $\left( -2,4\right) ,\,f\cap ,$v $x=4$ m\'{a} $f$ inflexn\'{y} bod, $f\left( 4\right) =\frac{10}{% 9}.$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =0,\,\lim_{x\longrightarrow \infty }f\left( x\right) =0.$ Asymptota v $% -\infty :y=0,$ asymptota v $\infty :y=0.\,H\left( f\right) =\left( -\infty ,% \frac{5}{4}\right\rangle $% \par Graf \ $f\left( x\right) =\frac{10x}{\left( x+2\right) ^{2}}$\FRAME{dtbpFX}{% 3.3754in}{2.2502in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.3754in;height 2.2502in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-52.3210912713316";yviewmax "1.30351462170269";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\frac{10x}{\left( x+2\right) ^{2}}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";function \TEXUX{$0$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU46MA02.wmf';tempfile-properties "XPR";}% }}$ \textbf{Pr\'{\i}klad 9. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\cos x+\ln \left( \cos x\right) .% \CustomNote{Prob_Solv_Hint}{% {}$D\left( f\right) =\bigcup_{k\in \mathbf{Z}}\left( -\frac{\pi }{2}+2k\pi ,% \frac{\pi }{2}+2k\pi \right) ,$ spojit\'{a}, nulov\'{e} body m\'{a}, ich v% \'{y}po\v{c}et je zlo\v{z}it\'{y}, mo\v{z}n\'{y} iba numericky, asymptota bez smernice $x=-\frac{\pi }{2}+2k\pi \,\,\left( \lim_{x\longrightarrow -% \frac{\pi }{2}^{+}}f\left( x\right) =-\infty \right) ,\,x=\frac{\pi }{2}% +2k\pi \,\,\left( \lim_{x\longrightarrow \frac{\pi }{2}^{-}}f\left( x\right) =-\infty \right) .$% \par Funkcia $f$ je p\'{a}rna \par Funkcia $f$ je periodick\'{a} s periodou $2\pi .$% \par $f\,\,^{\prime }\left( x\right) =\allowbreak -\sin x\left( 1+\frac{1}{\cos x}% \right) ,$ pre $x\in \left( -\frac{\pi }{2}+2k\pi ,0+2k\pi \right) ,\,f\nearrow ,$ pre $x\in \left( 0+2k\pi ,\frac{\pi }{2}+2k\pi \right) ,\,f\searrow ,$ $\max f\left( x\right) =f\left( 2k\pi \right) =1.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak -\left( \cos x+\frac{1}{% \cos ^{2}x}\right) ,$ pre $x\in \left( -\frac{\pi }{2}+2k\pi ,\frac{\pi }{2}% +2k\pi \right) ,\,f\cap ,$ nem\'{a} inflexn\'{y} bod. \par $\,H\left( f\right) =\left( -\infty ,1\right\rangle $% \par Graf \ $f\left( x\right) =\cos x+\ln \left( \cos x\right) $\FRAME{dtbpFX}{% 3.2915in}{2.1932in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2915in;height 2.1932in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5.01";xmax "5.01";xviewmin "-5.01";xviewmax "5.01";yviewmin "-10";yviewmax "1.021349";viewset"XY";rangeset"X";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\cos x+\ln \left( \cos x\right) $};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5.01,5.01";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";valid_file "T";tempfilename 'HU46NI03.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 10. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =16x\left( x-1\right) ^{3}.% \CustomNote{Prob_Solv_Hint}{% {}$D\left( f\right) =\mathbf{R},$ spojit\'{a}, $f\left( 0\right) =0,\,f\left( 1\right) =0$ asymptotu bez smernice nem\'{a}. \par Funkcia $f$ nie je nep\'{a}rna, funkcia $f$ nie je p\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak 16\left( x-1\right) ^{2}\left( 4x-1\right) ,$ pre $x\in \left( \frac{1}{4},\infty \right) ,\,f\nearrow ,$ pre $x\in \left( -\infty ,\frac{1}{4}\right) ,\,f\searrow ,$ $\min f\left( x\right) =f\left( \frac{1}{4}\right) =-\frac{27}{16}.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak 96\left( x-1\right) \left( 2x-1\right) ,$ pre $x\in \left( -\infty ,\frac{1}{2}\right) $ a $% \left( 1,\infty \right) ,\,f\cup ,$ pre $x\in \left( \frac{1}{2},1\right) ,\,f\cap ,$ v $x=\frac{1}{2},\,1$ m\'{a} $f$ inflexn\'{e} body, $f\left( \frac{1}{2}\right) =-1,\,f\left( 1\right) =0.$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =\infty ,\,\lim_{x\longrightarrow \infty }f\left( x\right) =\infty .$ Nem\'{a} asymptotu v $-\infty ,$ nem\'{a} asymptotu v $\infty .\,H\left( f\right) =\left\langle -\frac{27}{16},\infty \right) .$% \par Graf \ $f\left( x\right) =16x\left( x-1\right) ^{3}$\FRAME{dtbpFX}{3.1981in}{% 2.1318in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.1981in;height 2.1318in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-0.5";xmax "2";xviewmin "-0.5025";xviewmax "2.0025";yviewmin "-1.72016728348684";yviewmax "32.0336864808027";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$16x\left( x-1\right) ^{3}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-0.5,2";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'HU2OAI0V.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 11. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\frac{1+\ln x}{x}.% \CustomNote{Prob_Solv_Hint}{% {}$D\left( f\right) =\left( 0,\infty \right) ,$ spojit\'{a}, $f\left( \frac{1% }{e}\right) =0,\,x=0$ je asymptota bez smernice. $\left( \lim_{x\longrightarrow 0^{+}}f\left( x\right) =-\infty .\right) $% \par Funkcia $f$ nie je nep\'{a}rna, funkcia $f$ nie je p\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak -\frac{\ln x}{x^{2}},$ pre $% x\in \left( 0,1\right) ,\,f\nearrow ,$ pre $x\in \left( 1,\infty \right) ,\,f\searrow ,$ $\max f\left( x\right) =f\left( 1\right) =1.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{-1+2\ln x}{x^{3}},$ pre $x\in \left( e^{\frac{1}{2}},\infty \right) ,\,f\cup ,$ pre $x\in \left( 0,e^{\frac{1}{2}}\right) ,\,f\cap ,$ v $x=e^{\frac{1}{2}}$ m\'{a} $f$ inflexn% \'{y} bod, $f\left( e^{\frac{1}{2}}\right) =\frac{3}{2\sqrt{e}}.$% \par $\lim_{x\longrightarrow \infty }f\left( x\right) =0.$ Asymptota v $\infty :y=0.\,H\left( f\right) =\left( -\infty ,1\right\rangle .$% \par Graf \ $f\left( x\right) =\frac{1+\ln x}{x}$\FRAME{dtbpFX}{3.4272in}{2.2848in% }{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.4272in;height 2.2848in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "0.1";xmax "10";xviewmin "-0.1101";xviewmax "10.0101";yviewmin "-13.0398767808704";yviewmax "1.01402585092994";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\frac{1+\ln x}{x}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "0.1,10";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";function \TEXUX{$0$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-0.1,10";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'HU2O9R0U.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 12. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =-1+\sqrt{-x^{2}-4x+5}.% \CustomNote{Prob_Solv_Hint}{% {}$D\left( f\right) =\left\langle -5,1\right\rangle ,$ spojit\'{a}, $f\left( -2-2\sqrt{2}\right) =0,\,f\left( -2+2\sqrt{2}\right) =0,$ nem\'{a} asymptotu bez smernice. \par Funkcia $f$ nie je nep\'{a}rna, funkcia $f$ nie je p\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak -\frac{x+2}{\sqrt{-x^{2}-4x+5}}% , $ pre $x\in \left( -5,-2\right) ,\,f\nearrow ,$ pre $x\in \left( -2,1\right) ,\,f\searrow ,$ $\max f\left( x\right) =f\left( -2\right) =2,\,\min f\left( x\right) =f\left( -5\right) =f\left( 1\right) =-1.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{-9}{\left( -x^{2}-4x+5\right) ^{\frac{3}{2}}},$ pre $x\in \left( -5,1\right) ,\,f\cap ,$ nem\'{a} inflexn\'{y} bod. \par $\,H\left( f\right) =\left\langle -1,2\right\rangle .$% \par Graf \ $f\left( x\right) =-1+\sqrt{-x^{2}-4x+5}$\FRAME{dtbpFX}{3.1773in}{% 2.1179in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.1773in;height 2.1179in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "1";xviewmin "-5.006";xviewmax "1.006";yviewmin "-1.00299984695049";yviewmax "2.00284679743887";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$-1+\sqrt{-x^{2}-4x+5}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5,1";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'HU2O8R0T.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 13. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =x\ln \left( x^{2}\right) .% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R\setminus }\left\{ 0\right\} ,$ spojit\'{a}, $f\left( -1\right) =0,\,f\left( 1\right) =0,$ asymptoty bez smernice nem\'{a}$\,\,\left( \lim_{x\longrightarrow 0}f\left( x\right) =0\right) .$% \par Funkcia $f$ je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak 2+\ln \left( x^{2}\right) ,$ pre $x\in \left( -\infty ,-\frac{1}{e}\right) $ a $\left( \frac{1}{e},\infty \right) ,\,f\nearrow ,$ pre $x\in \left( -\frac{1}{e},0\right) $ a $\left( 0,% \frac{1}{e}\right) ,\,f\searrow ,$ $\limfunc{lokmin}f\left( x\right) =f\left( \frac{1}{e}\right) =-\frac{2}{e},\limfunc{lokmax}f\left( x\right) =f\left( -\frac{1}{e}\right) =\frac{2}{e}\,.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{2}{x},$ pre $x\in \left( 0,\infty \right) ,\,f\cup ,$ pre $x\in \left( -\infty ,0\right) ,\,f\cap ,$ nem\'{a} inflexn\'{y} bod. \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =\infty ,\,\lim_{x\longrightarrow \infty }f\left( x\right) =\infty .$ Nem\'{a} asymptoty v $-\infty ,\,\infty .\,H\left( f\right) =\left( -\infty ,\infty \right) .$% \par Graf \ $f\left( x\right) =x\ln \left( x^{2}\right) $\FRAME{dtbpFX}{3.3018in}{% 2.2018in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.3018in;height 2.2018in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-3.5";xmax "3.5";xviewmin "-3.507";xviewmax "3.507";yviewmin "-8.78687946102651";yviewmax "8.78687946102651";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$x\ln \left( x^{2}\right) $};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-3.5,3.5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'HU2O8B0S.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 14. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\frac{x^{4}}{\left( x+1\right) ^{3}}.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R\setminus }\left\{ -1\right\} ,$ spojit\'{a}, $f\left( 0\right) =0,$ asymptota bez smernice $% x=-1\,\,\left( \lim_{x\longrightarrow -1^{-}}f\left( x\right) =-\infty ,\,\lim_{x\longrightarrow -1^{+}}f\left( x\right) =\infty \right) .$% \par Funkcia $f$ nie je p\'{a}rna, $f$ nie je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \frac{x^{3}\left( x+4\right) }{% \left( x+1\right) ^{4}},$ pre $x\in \left( -\infty ,-4\right) $ a $\left( 0,\infty \right) ,\,f\nearrow ,$ pre $x\in \left( -4,-1\right) $ a $\left( -1,0\right) ,\,f\searrow ,$ $\limfunc{lokmax}f\left( x\right) =f\left( -4\right) =-\frac{256}{27},\,\limfunc{lokmin}f\left( x\right) =f\left( 0\right) =0.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{12x^{2}}{\left( 1+x\right) ^{5}},$ pre $x\in \left( -1,0\right) $ a $\left( 0,\infty \right) ,\,f\cup ,$ pre $x\in \left( -\infty ,-1\right) ,\,f\cap ,$ nem\'{a} inflexn% \'{y} bod.. \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =-\infty ,\,\lim_{x\longrightarrow \infty }f\left( x\right) =\infty .$ Asymptota v $% -\infty :y=x-3,$ asymptota v $\infty :y=x-3.\,H\left( f\right) =\left( -\infty ,-\frac{256}{27}\right\rangle \cup \left\langle 0,\infty \right) .$% \par Graf \ $f\left( x\right) =\frac{x^{4}}{\left( x+1\right) ^{3}}$\FRAME{dtbpFX% }{3.25in}{2.1664in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.25in;height 2.1664in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-53.536053243178";yviewmax "46.6639467475748";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\frac{x^{4}}{\left( x+1\right) ^{3}}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";function \TEXUX{$x-3$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU46PB04.wmf';tempfile-properties "XPR";}% }}$ \textbf{Pr\'{\i}klad 15. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\ln \frac{x+1}{x-1}.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\left( -\infty ,-1\right) \cup \left( 1,\infty \right) ,$ spojit\'{a}, asymptoty bez smernice $% x=-1,\,x=1\,\,\left( \lim_{x\longrightarrow -1^{-}}f\left( x\right) =-\infty ,\,\lim_{x\longrightarrow 1^{+}}f\left( x\right) =\infty \right) .$% \par Funkcia $f$ je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \frac{2}{1-x^{2}}$ pre $x\in \left( -\infty ,-1\right) $ a $\left( 1,\infty \right) ,\,f\searrow ,$nem% \'{a} lok\'{a}lne extr\'{e}my. \par $f\,\,^{\prime \prime }\left( x\right) =\frac{4x}{\left( x^{2}-1\right) ^{2}} $ pre $x\in \left( 1,\infty \right) ,\,f\cup ,$ pre $x\in \left( -\infty ,-1\right) ,\,f\cap ,$ nem\'{a} inflexn\'{y} bod.. \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =0,\,\lim_{x\longrightarrow \infty }f\left( x\right) =0.$ Asymptota v $% -\infty :y=0,$ asymptota v $\infty :y=0.\,H\left( f\right) =\left( -\infty ,0\right) \cup \left( 0,\infty \right) .$% \par Graf \ $f\left( x\right) =\ln \frac{x+1}{x-1}$\FRAME{dtbpFX}{3.3537in}{% 2.2364in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.3537in;height 2.2364in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-2.5";xmax "2.5";xviewmin "-2.505";xviewmax "2.505";yviewmin "-41.9539665231605";yviewmax "35.6545578230458";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\ln \frac{x+1}{x-1}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-2.5,2.5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";function \TEXUX{$0$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-2.5,2.5";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU2OMM16.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 16. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\left\vert x^{2}-4x+1\right\vert .% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R},$ spojit\'{a}, $% f\left( 2-\sqrt{3}\right) =0,\,f\left( 2+\sqrt{3}\right) =0$ asymptotu bez smernice nem\'{a}. \par Funkcia $f$ nie je nep\'{a}rna, funkcia $f$ nie je p\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \left\{ \begin{tabular}{cc} $2x-4$ & $x\in \left( -\infty ,2-\sqrt{3}\right) \cup \left( 2+\sqrt{3}% ,\infty \right) $ \\ $-2x+4$ & $x\in \left( 2-\sqrt{3},2+\sqrt{3}\right) $% \end{tabular}% \right. ,$ pre $x\in \left( 2-\sqrt{3},2\right) $ a $\left( 2+\sqrt{3}% ,\infty \right) ,\,f\nearrow ,$ pre $x\in \left( -\infty ,2-\sqrt{3}\right) $ a $\left( 2,2+\sqrt{3}\right) ,\,f\searrow ,$ $\limfunc{lokmax}f\left( x\right) =f\left( 2\right) =3,\,\min f\left( x\right) =f\left( 2-\sqrt{3}% \right) =f\left( 2+\sqrt{3}\right) =0.\left( f\,\,^{\prime }\left( 2-\sqrt{3}% \right) ,\,f\,\,^{\prime }\left( 2+\sqrt{3}\right) \nexists \right) .$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \left\{ \begin{tabular}{cc} $2$ & $x\in \left( -\infty ,2-\sqrt{3}\right) \cup \left( 2+\sqrt{3},\infty \right) $ \\ $-2$ & $x\in \left( 2-\sqrt{3},2+\sqrt{3}\right) $% \end{tabular}% \right. ,$ pre $x\in \left( -\infty ,2-\sqrt{3}\right) $ a $\left( 2+\sqrt{3}% ,\infty \right) ,\,f\cup ,$ pre $x\in \left( 2-\sqrt{3},2+\sqrt{3}\right) ,\,f\cap ,$ nem\'{a} inflexn\'{e} body, $\left( \,f\,\,^{\prime \prime }\left( 2-\sqrt{3}\right) ,\,f\,\,^{\prime \prime }\left( 2+\sqrt{3}\right) \,\nexists \right) .$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =\infty ,\,\lim_{x\longrightarrow \infty }f\left( x\right) =\infty .$ Nem\'{a} asymptotu v $-\infty ,$ nem\'{a} asymptotu v $\infty .\,H\left( f\right) =\left\langle 0,\infty \right) .$% \par Graf \ $f\left( x\right) =\left\vert x^{2}-4x+1\right\vert $\FRAME{dtbpFX}{% 3.1981in}{2.1318in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.1981in;height 2.1318in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-3";xmax "7";xviewmin "-3.01";xviewmax "7.01";yviewmin "0.102805836139169";yviewmax "22.021875318845";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\left\vert x^{2}-4x+1\right\vert $};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-3,7";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'HU2ORI18.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 17. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =e^{-x^{2}}.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R},$ spojit\'{a}, nem% \'{a} nulov\'{y} bod, asymptotu bez smernice nem\'{a}. \par Funkcia $f$ je p\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak -2xe^{-x^{2}},$ pre $x\in \left( -\infty ,0\right) ,\,f\nearrow ,$ pre $x\in \left( 0,\infty \right) ,\,f\searrow ,$\thinspace $\max f\left( x\right) =f\left( 0\right) =1.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak 2\left( 2x^{2}-1\right) e^{-x^{2}},$ pre $x\in \left( -\infty ,-\frac{\sqrt{2}}{2}\right) $ a $% \left( \frac{\sqrt{2}}{2},\infty \right) ,\,f\cup ,$ pre $x\in \left( -\frac{% \sqrt{2}}{2},\frac{\sqrt{2}}{2}\right) ,\,f\cap ,$ v $x=-\frac{\sqrt{2}}{2},% \frac{\sqrt{2}}{2}$ m\'{a} $f$\ inflexn\'{e} body, $f\left( -\frac{\sqrt{2}}{% 2}\right) =\allowbreak f\left( \frac{\sqrt{2}}{2}\right) =e^{-\frac{1}{2}}.$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =0,\,\lim_{x\longrightarrow \infty }f\left( x\right) =0.$ Asymptota v $% -\infty :y=0,$ asymptota v $\infty :y=0.\,H\left( f\right) =\left( 0,1\right\rangle .$% \par Graf \ $f\left( x\right) =e^{-x^{2}}$\FRAME{dtbpFX}{3.25in}{2.1664in}{0pt}{}{% }{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.25in;height 2.1664in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-2.75";xmax "2.75";xviewmin "-2.7555";xviewmax "2.7555";yviewmin "-0.000999228692672";yviewmax "1.00022792136494";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$e^{-x^{2}}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-2.75,2.75";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";function \TEXUX{$0$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-2.75,2.75";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU2OUK1A.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 18. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\left( 1-3x\right) e^{2x}.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R},$ spojit\'{a}, $% f\left( \frac{1}{3}\right) =0$ asymptotu bez smernice nem\'{a}. \par Funkcia $f$ nie je nep\'{a}rna, funkcia $f$ nie je p\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \left( -1-6x\right) e^{2x},$ pre $x\in \left( -\infty ,-\frac{1}{6}\right) ,\,f\nearrow ,$ pre $x\in \left( -\frac{1}{6},\infty \right) ,\,f\searrow ,$\thinspace $\max f\left( x\right) =f\left( -\frac{1}{6}\right) =\frac{3}{2}e^{-\frac{1}{3}}.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak -4\left( 2+3x\right) e^{2x},$ pre $x\in \left( -\infty ,-\frac{2}{3}\right) ,\,f\cup ,$ pre $x\in \left( -\frac{2}{3},\infty \right) ,\,f\cap ,$ v $x=-\frac{2}{3}$ m\'{a} $f$% \ inflexn\'{y} bod, $f\left( -\frac{2}{3}\right) =\allowbreak 3e^{-\frac{4}{3% }}.$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =0,\,\lim_{x\longrightarrow \infty }f\left( x\right) =\infty .$ Asymptota v $% -\infty :y=0,$ nem\'{a} asymptotu v $\infty .\,H\left( f\right) =\left( -\infty ,\frac{3}{2}e^{-\frac{1}{3}}\right\rangle .$% \par Graf \ $f\left( x\right) =\left( 1-3x\right) e^{2x}$\FRAME{dtbpFX}{3.3754in}{% 2.2502in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.3754in;height 2.2502in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-2.5";xmax "2.5";xviewmin "-2.505";xviewmax "2.505";yviewmin "-20";yviewmax "2.040341";viewset"XY";rangeset"X";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\left( 1-3x\right) e^{2x}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-2.5,2.5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";function \TEXUX{$0$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-2.5,0.3";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'HU2OYR1C.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 19. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =xe^{x}.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R},$ spojit\'{a}, $% f\left( 0\right) =0,$ asymptotu bez smernice nem\'{a}. \par Funkcia $f$ nie je nep\'{a}rna, funkcia $f$ nie je p\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \left( 1+x\right) e^{x},$ pre $% x\in \left( -1,\infty \right) ,\,f\nearrow ,$ pre $x\in \left( -\infty ,-1\right) ,\,f\searrow ,$\thinspace $\min f\left( x\right) =f\left( -1\right) =-\frac{1}{e}.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \left( x+2\right) e^{x},$ pre $x\in \left( -2,\infty \right) ,\,f\cup ,$ pre $x\in \left( -\infty ,-2\right) ,\,f\cap ,$ v $x=-2$ m\'{a} $f$\ inflexn\'{y} bod, $f\left( -2\right) =\allowbreak -\frac{2}{e^{2}}.$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =0,\,\lim_{x\longrightarrow \infty }f\left( x\right) =\infty .$ Asymptota v $% -\infty :y=0,$ nem\'{a} asymptotu v $\infty .\,H\left( f\right) =\left\langle -\frac{1}{e},\infty \right) .$% \par Graf \ $f\left( x\right) =xe^{x}$\FRAME{dtbpFX}{3.2603in}{2.1733in}{0pt}{}{}{% Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2603in;height 2.1733in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-2.5";xmax "2.5";xviewmin "-2.505";xviewmax "2.505";yviewmin "-0.398661";yviewmax "15.04";viewset"XY";rangeset"X";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$xe^{x}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-2.5,2.5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";function \TEXUX{$0$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-2.5,0.3";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'HU2QVO1Y.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 20. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\frac{2}{e^{x}-3}.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R\setminus }\left\{ \ln 3\right\} ,$ spojit\'{a}, nem\'{a} nulov\'{y} bod, asymptota bez smernice $x=\ln 3\,\,\left( \lim_{x\longrightarrow \ln 3^{-}}f\left( x\right) =-\infty ,\,\lim_{x\longrightarrow \ln 3^{+}}f\left( x\right) =\infty \right) .$% \par Funkcia $f$ nie je p\'{a}rna, $f$ nie je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \frac{-2e^{x}}{\left( e^{x}-3\right) ^{2}},$ pre $x\in \left( -\infty ,\ln 3\right) $ a $\left( \ln 3,\infty \right) ,\,f\searrow ,$ nem\'{a} lok\'{a}lne extr\'{e}my. \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{2e^{2x}+6e^{x}}{% \left( e^{x}-3\right) ^{3}},$ pre $x\in \left( \ln 3,\infty \right) ,\,f\cup ,$ pre $x\in \left( -\infty ,\ln 3\right) ,\,f\cap ,$ nem\'{a} inflexn\'{y} bod. \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =-\frac{2}{3}% ,\,\lim_{x\longrightarrow \infty }f\left( x\right) =0.$ Asymptota v $-\infty :y=-\frac{2}{3},$ asymptota v $\infty :y=0.\,H\left( f\right) =\left( -\infty ,-\frac{2}{3}\right) \cup \left( 0,\infty \right) .$% \par Graf \ $f\left( x\right) =\frac{2}{e^{x}-3}$\FRAME{dtbpFX}{3.2707in}{2.1802in% }{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2707in;height 2.1802in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-1";xmax "3";xviewmin "-1.004";xviewmax "3.004";yviewmin "-20.3613927944314";yviewmax "19.71860719254";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\frac{2}{e^{x}-3}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-1,3";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";function \TEXUX{$0$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0.3,3";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"X";function \TEXUX{$-\frac{2}{3}$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-1,0.3";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'HU2PDU1G.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 21. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\left( \frac{1-x}{1+x}\right) ^{2}.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R\setminus }\left\{ -1\right\} ,$ spojit\'{a}, $f\left( 1\right) =0,$ asymptota bez smernice $% x=-1\,\,\left( \lim_{x\longrightarrow -1}f\left( x\right) =\infty \right) .$% \par Funkcia $f$ nie je p\'{a}rna, $f$ nie je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak 4\frac{x-1}{\left( 1+x\right) ^{3}}$ pre $x\in \left( -\infty ,-1\right) $ a $\left( 1,\infty \right) ,\,f\nearrow ,$ pre $x\in \left( -1,1\right) ,\,f\searrow ,$ $\min f\left( x\right) =f\left( 1\right) =0.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak 8\frac{2-x}{\left( 1+x\right) ^{4}}$ pre $x\in \left( -\infty ,-1\right) $ a $\left( -1,2\right) ,\,f\cup ,$ pre $x\in \left( 2,\infty \right) ,\,f\cap ,$ v $x=2$ m\'{a} $f$\ inflexn\'{y} bod, $f\left( 2\right) =\frac{1}{9}$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =1,\,\lim_{x\longrightarrow \infty }f\left( x\right) =1.$ Asymptota v $% -\infty :y=1,$ asymptota v $\infty :y=1.\,H\left( f\right) =\left\langle 0,\infty \right) .$% \par Graf \ $f\left( x\right) =\left( \frac{1-x}{1+x}\right) ^{2}$\FRAME{dtbpFX}{% 3.2707in}{2.1802in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2707in;height 2.1802in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-0.051315978965186";yviewmax "51.3985694038325";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\left( \frac{1-x}{1+x}\right) ^{2}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";function \TEXUX{$1$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU2PXL1J.wmf';tempfile-properties "XPR";}% }}$ \textbf{Pr\'{\i}klad 22. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =x-\left\vert \sin x\right\vert .% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R},$ spojit\'{a}, $% f\left( 0\right) =0,$ nem\'{a} asymptotu bez smernice \par Funkcia $f$ nie je p\'{a}rna, $f$ nie je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\left\{ \begin{tabular}{cc} $1-\cos x$ & $x\in \left( 2k\pi ,\left( 2k+1\right) \pi \right) $ \\ $1+\cos x$ & $x\in \left( \left( 2k+1\right) \pi ,\left( 2k+2\right) \pi \right) $% \end{tabular}% \right. ,\,$% \par $k\in \mathbf{Z},$ pre $x\in \left( 2k\pi ,\left( 2k+1\right) \pi \right) $ a $\left( \left( 2k+1\right) \pi ,\left( 2k+2\right) \pi \right) ,\,f\nearrow ,$ $f\,\,^{\prime }\left( 2k\pi \right) ,\,f\,\,^{\prime }\left( \left( 2k+1\right) \pi \right) $ $\nexists ,k\in \mathbf{Z},$ nem% \'{a} lok\'{a}lne extr\'{e}my. \par $f\,\,^{\prime \prime }\left( x\right) =\left\{ \begin{tabular}{cc} $\sin x$ & $x\in \left( 2k\pi ,\left( 2k+1\right) \pi \right) $ \\ $-\sin x$ & $x\in \left( \left( 2k+1\right) \pi ,\left( 2k+2\right) \pi \right) $% \end{tabular}% \right. ,\,$% \par $k\in \mathbf{Z},$ pre $x\in \left( 2k\pi ,\left( 2k+1\right) \pi \right) $ \ a $\ \left( \left( 2k+1\right) \pi ,\left( 2k+2\right) \pi \right) ,\,\,f\cup ,$ $f\,\,^{\prime \prime }\left( 2k\pi \right) ,\,f\,\,^{\prime \prime }\left( \left( 2k+1\right) \pi \right) $ $\nexists ,k\in \mathbf{Z},$ nem\'{a} inflexn\'{e} body. \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =-\infty ,\,\lim_{x\longrightarrow \infty }f\left( x\right) =\infty .$ Nem\'{a} asymptotu v $-\infty ,$ nem\'{a} asymptotu v $\infty .\,H\left( f\right) =% \mathbf{R}.$% \par Graf \ $f\left( x\right) =x-\left\vert \sin x\right\vert $\FRAME{dtbpFX}{% 3.2707in}{2.1802in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2707in;height 2.1802in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-5.96892427466314";yviewmax "4.05107572533686";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$x-\left\vert \sin x\right\vert $};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";valid_file "T";tempfilename 'HU2Q1K1L.wmf';tempfile-properties "XPR";}% }}$ \textbf{Pr\'{\i}klad 23. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\sin x+\cos x.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R},$ spojit\'{a}, $% f\left( \frac{3\pi }{4}+2k\pi \right) =f\left( \frac{7\pi }{4}+2k\pi \right) =0,$ nem\'{a} asymptotu bez smernice \par Funkcia $f$ nie je p\'{a}rna, $f$ nie je nep\'{a}rna \par Funkcia $f$ je periodick\'{a} s periodou $2\pi .$ Budeme ju vy\v{s}etrova% \v{t} na intervale $\left\langle 0,2\pi \right\rangle $% \par $f\,\,^{\prime }\left( x\right) =\allowbreak \cos x-\sin x,$ pre $x\in \left( 0,\frac{\pi }{4}\right) $ a $\left( \frac{5\pi }{4},2\pi \right) ,\,f\nearrow ,$ pre $x\in \left( \frac{\pi }{4},\frac{5\pi }{4}\right) ,\,f\searrow ,$ $\limfunc{lokmax}f\left( x\right) =f\left( \frac{\pi }{4}% \right) =\sqrt{2},\,\limfunc{lokmin}f\left( x\right) =f\left( \frac{5\pi }{4}% \right) =-\sqrt{2}.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak -\sin x-\cos x,\,$pre $% x\in \left( \frac{3\pi }{4},\frac{7\pi }{4}\right) ,\,\,f\cup ,$ pre $x\in \left( 0,\frac{3\pi }{4}\right) $ a $\left( \frac{7\pi }{4},2\pi \right) ,\,\,f\cap ,$ v $x=\frac{3\pi }{4},\,x=\frac{7\pi }{4}\ $ m\'{a} $f$\ inflexn% \'{e} body, $f\left( \frac{3\pi }{4}\right) =0,\,f\left( \frac{7\pi }{4}% \right) =0.$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) \nexists ,\,\lim_{x\longrightarrow \infty }f\left( x\right) \nexists .$ Nem\'{a} asymptotu v $-\infty ,$ nem\'{a} asymptotu v $\infty .\,H\left( f\right) =\left\langle -\sqrt{2},\sqrt{2}\right\rangle .$% \par Graf \ $f\left( x\right) =\sin x+\cos x$\FRAME{dtbpFX}{3.2085in}{2.1387in}{% 0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2085in;height 2.1387in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-1.4168236160782";yviewmax "1.41649389822316";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\sin x+\cos x$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";valid_file "T";tempfilename 'HU2Q441N.wmf';tempfile-properties "XPR";}% }}$ \textbf{Pr\'{\i}klad 24. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\frac{x}{1+x^{2}}.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R},$ spojit\'{a}, $% f\left( 0\right) =0,$ asymptotu bez smernice nem\'{a}. \par Funkcia $f$ je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \frac{1-x^{2}}{\left( 1+x^{2}\right) ^{2}},$ pre $x\in \left( -1,1\right) ,\,f\nearrow ,$ pre $% x\in \left( -\infty ,-1\right) $ a $\left( 1,\infty \right) ,\,f\searrow ,$ $% \max f\left( x\right) =f\left( 1\right) =\frac{1}{2},\min f\left( x\right) =f\left( -1\right) =-\frac{1}{2}\,.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{2x\left( x^{2}-3\right) }{\left( 1+x^{2}\right) ^{3}},$ pre $x\in \left( -\sqrt{3}% ,0\right) $ a $\left( \sqrt{3},\infty \right) ,\,f\cup ,$ pre $x\in \left( -\infty ,-\sqrt{3}\right) $ a $\left( 0,\sqrt{3}\right) ,\,f\cap ,$ v $x=-% \sqrt{3},\,0,\,\sqrt{3}$ m\'{a} $f$\ inflexn\'{e} body, $f\left( 0\right) =0,\,f\left( -\sqrt{3}\right) =-\frac{\sqrt{3}}{4},f\left( \sqrt{3}\right) =% \frac{\sqrt{3}}{4}\,.$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =0,\,\lim_{x\longrightarrow \infty }f\left( x\right) =0.$ Asymptota v $% -\infty :y=0,$ asymptota v $\infty :y=0.\,H\left( f\right) =\left\langle -% \frac{1}{2},\frac{1}{2}\right\rangle .$% \par Graf \ $f\left( x\right) =\frac{x}{1+x^{2}}$\FRAME{dtbpFX}{3.3018in}{2.2018in% }{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.3018in;height 2.2018in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-10";xmax "10";xviewmin "-10.02";xviewmax "10.02";yviewmin "-0.498733031674208";yviewmax "0.498733031674208";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\frac{x}{1+x^{2}}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-10,10";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";function \TEXUX{$0$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-10,10";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU46SF07.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 25. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\ln \left( \cos x\right) .% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\bigcup_{k\in \mathbf{Z}% }\left( -\frac{\pi }{2}+2k\pi ,\frac{\pi }{2}+2k\pi \right) ,$ spojit\'{a}, $% f\left( 0+2k\pi \right) =0,$ asymptota bez smernice $x=-\frac{\pi }{2}+2k\pi \,\,\left( \lim_{x\longrightarrow -\frac{\pi }{2}^{+}}f\left( x\right) =-\infty \right) ,\,x=\frac{\pi }{2}+2k\pi \,\,\left( \lim_{x\longrightarrow \frac{\pi }{2}^{-}}f\left( x\right) =-\infty \right) .$% \par Funkcia $f$ je p\'{a}rna \par Funkcia $f$ je periodick\'{a} s periodou $2\pi .$% \par $f\,\,^{\prime }\left( x\right) =\allowbreak -\frac{\sin x}{\cos x},$ pre $% x\in \left( -\frac{\pi }{2}+2k\pi ,0+2k\pi \right) ,\,f\nearrow ,$ pre $x\in \left( 0+2k\pi ,\frac{\pi }{2}+2k\pi \right) ,\,f\searrow ,$ $\max f\left( x\right) =f\left( 2k\pi \right) =0.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak -\frac{1}{\cos ^{2}x},$ pre $x\in \left( -\frac{\pi }{2}+2k\pi ,\frac{\pi }{2}+2k\pi \right) ,\,f\cap ,$ nem\'{a} inflexn\'{y} bod. \par $\,H\left( f\right) =\left( -\infty ,0\right\rangle $% \par Graf \ $f\left( x\right) =\ln \left( \cos x\right) $\FRAME{dtbpFX}{3.2396in}{% 2.1594in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2396in;height 2.1594in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-7.5";xmax "7.5";xviewmin "-8.5";xviewmax "8.5";yviewmin "-10.42781";yviewmax "0.020378";viewset"XY";rangeset"X";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\ln \left( \cos x\right) $};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-7.5,7.5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'HU2QCV1R.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 26. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\ln \left\vert x^{2}-5x+6\right\vert .% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R\setminus }\left\{ 2,\,3\right\} ,$ spojit\'{a}, $f\left( \frac{5-\sqrt{5}}{2}\right) =0,\,f\left( \frac{5+\sqrt{5}}{2}\right) =0,$ asymptoty bez smernice $% x=2,\,x=3\,$ $\,\,\left( \lim_{x\longrightarrow 2}f\left( x\right) =-\infty ,\,\lim_{x\longrightarrow 3}f\left( x\right) =-\infty \right) .$% \par Funkcia $f$ nie je nep\'{a}rna, funkcia $f$ nie je p\'{a}rna. \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \frac{2x-5}{x^{2}-5x+6},$ pre $% x\in \left( 2,\frac{5}{2}\right) $ a $\left( 3,\infty \right) ,\,f\nearrow ,$ pre $x\in \left( -\infty ,2\right) $ a $\left( \frac{5}{2},3\right) ,\,f\searrow ,$ $\limfunc{lokmax}f\left( x\right) =f\left( \frac{5}{2}% \right) =-\ln 4.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{-2x^{2}+10x-13}{% \left( x^{2}-5x+6\right) ^{2}},$ pre $x\in \left( -\infty ,2\right) $ a $% \left( 2,3\right) $ a $\left( 3,\infty \right) ,\,f\cap ,$ nem\'{a} inflexn% \'{y} bod. \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =\infty ,\,\lim_{x\longrightarrow \infty }f\left( x\right) =\infty .$ Nem\'{a} asymptoty v $-\infty ,\,\infty .\,H\left( f\right) =\mathbf{R}.$% \par Graf \ $f\left( x\right) =\ln \left\vert x^{2}-5x+6\right\vert $\FRAME{dtbpFX% }{3.1877in}{2.1248in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.1877in;height 2.1248in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-20.3487615524293";yviewmax "4.04970145421483";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\ln \left\vert x^{2}-5x+6\right\vert $};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";valid_file "T";tempfilename 'HU2QKA1U.wmf';tempfile-properties "XPR";}% }}$ \textbf{Pr\'{\i}klad 27. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\frac{x}{2}+\func{arctg}x.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R},$ spojit\'{a}, $% f\left( 0\right) =0,\,$asymptoty bez smernice nem\'{a}. \par Funkcia $f$ je nep\'{a}rna. \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \frac{3+x^{2}}{2\left( 1+x^{2}\right) },$ pre $x\in \mathbf{R},\,f\nearrow ,$ nem\'{a} lok\'{a}lne extr\'{e}my. \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{-2x}{\left( 1+x^{2}\right) ^{2}},$ pre $x\in \left( -\infty ,0\right) ,\,f\cup $ $\ $pre $x\in \left( 0,\infty \right) ,\,f\cap ,$ v $x=0$ m\'{a} $f$\ inflexn\'{y} bod, $f\left( 0\right) =0.$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =-\infty ,\,\lim_{x\longrightarrow \infty }f\left( x\right) =\infty .$ Asymptota v $% -\infty :y=\frac{x}{2}-\frac{\pi }{2},$ asymptota v bode $\,\infty :y=\frac{x% }{2}+\frac{\pi }{2}.\,H\left( f\right) =\mathbf{R}.$% \par Graf \ $f\left( x\right) =\frac{x}{2}+\func{arctg}x.\frac{x}{2}+\arctan x$% \FRAME{dtbpFX}{3.2396in}{2.1594in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2396in;height 2.1594in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-4.07893791944849";yviewmax "4.07893791944849";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\frac{x}{2}+\arctan x$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";function \TEXUX{$\frac{x}{2}-\frac{\pi }{2}$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-5,0.5";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"X";function \TEXUX{$\frac{x}{2}+\frac{\pi }{2}$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-0.5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'HU2QN71W.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 28. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\frac{1}{1-x^{2}}.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R\setminus }\left\{ -1,1\right\} ,$ spojit\'{a}, nem\'{a} nulov\'{e} body, asymptota bez smernice $x=-1\,\,\left( \lim_{x\longrightarrow -1^{-}}f\left( x\right) =-\infty ,\,\lim_{x\longrightarrow -1^{+}}f\left( x\right) =\infty \right) ,\,x=1\,\,$% \par $\left( \lim_{x\longrightarrow 1^{-}}f\left( x\right) =\infty ,\,\lim_{x\longrightarrow -1^{+}}f\left( x\right) =-\infty \right) .$% \par Funkcia $f$ je p\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\allowbreak \frac{2x}{\left( 1-x^{2}\right) ^{2}}$ pre $x\in \left( 0,1\right) $ a $\left( 1,\infty \right) ,\,f\nearrow ,$ pre $x\in \left( -\infty ,-1\right) $ a $\ \left( -1,0\right) ,\,f\searrow ,$ $\limfunc{lokmin}f\left( x\right) =f\left( 0\right) =1.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \frac{6x^{2}+2}{\left( 1-x^{2}\right) ^{3}}$ pre $x\in \left( -1,1\right) ,\,f\cup ,$ pre $x\in \left( -\infty ,-1\right) $ a $\left( 1,\infty \right) ,\,f\cap ,$ nem\'{a} inflexn\'{y} bod. \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =0,\,\lim_{x\longrightarrow \infty }f\left( x\right) =0.$ Asymptota v $% -\infty :y=0,$ asymptota v $\infty :y=0.\,H\left( f\right) =\left( -\infty ,0\right) \cup \left( 1,\infty \right) .$% \par Graf \ $f\left( x\right) =\frac{1}{1-x^{2}}$\FRAME{dtbpFX}{3.3018in}{2.2018in% }{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.3018in;height 2.2018in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-49.6208334240566";yviewmax "50.5791662908599";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\frac{1}{1-x^{2}}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";function \TEXUX{$0$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU46U108.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 29. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =\frac{1}{\left\vert x^{2}-9\right\vert }.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R\setminus }\left\{ -3,3\right\} ,$ spojit\'{a}, nem\'{a} nulov\'{e} body, asymptota bez smernice $x=-3\,\,\left( \lim_{x\longrightarrow -3}f\left( x\right) =\infty \right) ,\,x=3\,\,\left( \lim_{x\longrightarrow 3}f\left( x\right) =\infty \right) .$% \par Funkcia $f$ je p\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\left\{ \begin{tabular}{cc} $\frac{-2x}{\left( x^{2}-9\right) ^{2}}$ & $x\in \left( -\infty ,-3\right) \cup \left( 3,\infty \right) $ \\ $\frac{2x}{\left( 9-x^{2}\right) ^{2}}$ & $x\in \left( -3,3\right) $% \end{tabular}% \right. ,\,$pre $x\in \left( -\infty ,-3\right) $ a $\left( 0,3\right) ,\,f\nearrow ,$ pre $x\in \left( -3,0\right) $ a $\left( 3,\infty \right) ,\,f\searrow ,$ $\limfunc{lokmin}f\left( x\right) =f\left( 0\right) =\frac{1% }{9}.$% \par $f\,\,^{\prime \prime }\left( x\right) =\left\{ \begin{tabular}{cc} $\frac{18+6x^{2}}{\left( x^{2}-9\right) ^{3}}$ & $x\in \left( -\infty ,-3\right) \cup \left( 3,\infty \right) $ \\ $\frac{18+6x^{2}}{\left( 9-x^{2}\right) ^{3}}$ & $x\in \left( -3,3\right) $% \end{tabular}% \right. ,\,$pre $x\in \left( -\infty ,-3\right) $ a $\left( -3,3\right) $ a $% \left( 3,\infty \right) ,\,f\cup ,$ nem\'{a} inflexn\'{y} bod. \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =0,\,\lim_{x\longrightarrow \infty }f\left( x\right) =0.$ Asymptota v $% -\infty :y=0,$ asymptota v $\infty :y=0.\,H\left( f\right) =\left( 0,\infty \right) .$% \par Graf \ $f\left( x\right) =\frac{1}{\left\vert x^{2}-9\right\vert }$\FRAME{% dtbpFX}{3.25in}{2.1664in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.25in;height 2.1664in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5.01";xmax "5.01";xviewmin "-5.01";xviewmax "5.01";yviewmin "-0.050087";yviewmax "13.13689";viewset"XY";rangeset"X";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\frac{1}{\left\vert x^{2}-9\right\vert }$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5.01,5.01";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";function \TEXUX{$0$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-5.01,5.01";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU46V409.wmf';tempfile-properties "XPR";}}}$ \textbf{Pr\'{\i}klad 30. }Zistite priebeh funkcie a na\v{c}rtnite jej graf ak $f\left( x\right) =xe^{-\frac{x^{2}}{2}}.% \CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\mathbf{R},$ spojit\'{a}, $% f\left( 0\right) =0,$ nem\'{a} asymptotu bez smernice. \par Funkcia $f$ je nep\'{a}rna \par Funkcia $f$ nie je periodick\'{a} \par $f\,\,^{\prime }\left( x\right) =\left( \allowbreak 1-x^{2}\right) e^{-\frac{% x^{2}}{2}}$pre $x\in \left( -1,1\right) ,\,f\nearrow ,$ pre $x\in \left( -\infty ,-1\right) $ a $\left( 1,\infty \right) ,\,f\searrow ,$ $\max f\left( x\right) =f\left( 1\right) =e^{-\frac{1}{2}},\,\min f\left( x\right) =f\left( -1\right) =-e^{-\frac{1}{2}}.$% \par $f\,\,^{\prime \prime }\left( x\right) =\allowbreak \left( \allowbreak x^{3}-3x\right) e^{-\frac{x^{2}}{2}},\,$pre $x\in \left( -\sqrt{3},0\right) $ a $\left( \sqrt{3},\infty \right) ,\,f\cup ,$ pre $x\in \left( -\infty ,-% \sqrt{3}\right) $ a $\left( 0,\sqrt{3}\right) ,\,f\cap ,$ v $x=-\sqrt{3}% ,\,0,\,\sqrt{3}$\ m\'{a} $f$ inflexn\'{e} body, $f\left( -\sqrt{3}\right) =-% \sqrt{3}e^{-\frac{3}{2}},\,\,f\left( 0\right) =0,\,f\left( \sqrt{3}\right) =% \sqrt{3}e^{-\frac{3}{2}}$% \par $\lim_{x\longrightarrow -\infty }f\left( x\right) =0,\,\lim_{x\longrightarrow \infty }f\left( x\right) =0.$ Asymptota v $% -\infty :y=0,$ asymptota v $\infty :y=0.\,H\left( f\right) =\left\langle -e^{-\frac{1}{2}},e^{-\frac{1}{2}}\right\rangle .$% \par Graf \ $f\left( x\right) =xe^{-\frac{x^{2}}{2}}$\FRAME{dtbpFX}{3.2188in}{% 2.1456in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.2188in;height 2.1456in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin "-0.606738639117682";yviewmax "0.606738639117682";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$xe^{-\frac{x^{2}}{2}}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";function \TEXUX{$0$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";valid_file "T";tempfilename 'HU2R9524.wmf';tempfile-properties "XPR";}% }}$ \begin{center} \begin{tabular}{|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Zbierka}{}{}{Z.tex}}% %BeginExpansion \msihyperref{Zbierka}{}{}{Z.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za I} \section{Zbierka \'{u}loh} \end{document}