\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Saturday, June 01, 2002 20:04:40} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za I online - D\U{f4}kazy\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{Aplik\'{a}cie deriv\'{a}ci\'{\i}} \subsection{D\^{o}kaz nutnej a posta\v{c}uj\'{u}cej podmienky konvexnosti} \textbf{D\^{o}kaz: }Vetu dok\'{a}\v{z}eme pre \emph{konvexnos\v{t}}, \emph{% konk\'{a}vnos\v{t}} sa dokazuje podobne. $\left( \Longleftarrow \right) $ Nech $f\,$\/$\,^{\prime \prime }(x)>0$ na intervale $I$ a $c\in I$ je vn\'{u}torn\'{y} bod. Potom je $f\,$\/$% \,^{\prime }$ rast\'{u}ca funkcia vn\'{u}tri intervalu $I$. Pre pevn\'{e} $% x\in I,\,x\neq c$ potom existuje $z\in \left\langle x,c\right\rangle $ (alebo $z\in \left\langle c,x\right\rangle $) tak\'{e}, \v{z}e \[ f(x)-f(c)=f\,\/\,^{\prime }(z)(x-c), \]% potom ak dosad\'{\i}me do defin\'{\i}cie \hyperref{funkcie}{}{}{M66.tex#2} $% g\left( x\right) $ dostaneme \[ g(x)=[f\,\/\,^{\prime }(z)-f\,\/\,^{\prime }(c)](x-c). \]% Preto\v{z}e $f\,$\/$\,^{\prime }$ je rast\'{u}ca a bod $z$ le\v{z}\'{\i} medzi bodmi $x$ a $c$, potom $g(x)>0,\,\forall x$ z vn\'{u}tra $I,\,x\neq c$% . Teda $l_{c}$ le\v{z}\'{\i} pod grafom funkcie $f$ (s v\'{y}nimkou bodu dotyku) a tak $f$ je konvexn\'{a} funkcia. $\left( \Longrightarrow \right) $ Nech $f$ je konvexn\'{a} na $I$ a $c\in I$ je vn\'{u}torn\'{y} bod. Potom \[ \,g\left( u\right) =f\left( u\right) -f\left( c\right) -f\,\/\,^{\prime }\left( c\right) \left( u-c\right) >0,\,\forall u\in I\setminus \left\{ c\right\} . \]% Preto\v{z}e $f$ je dvakr\'{a}t diferencovate\v{l}n\'{a} vo vn\'{u}tri intervalu $I$, pre pevn\'{e} $u\in I,\,u\neq c$ potom existuje $z\in \left( u,c\right) $ (alebo $z\in \left( c,u\right) $) tak\'{e}, \v{z}e $% f(u)-f(c)=f\,$\/$\,^{\prime }(z)(u-c),$ potom ak dosad\'{\i}me do funkcie $g$ dostaneme: $g(u)=[f\,$\/$\,^{\prime }(z)-f\,$\/$\,^{\prime }(c)](u-c)>0.$% Preto\v{z}e $f$ je dvakr\'{a}t diferencovate\v{l}n\'{a} vo vn\'{u}tri intervalu $I$, pre pevn\'{e} $z\in I,\,z\neq c$ potom existuje $x\in \left( z,c\right) $ (alebo $x\in \left( c,z\right) $) tak\'{e}, \v{z}e $f\,$\/$% \,^{\prime }(z)-f\,$\/$\,^{\prime }(c)=f\,$\/$\,^{\prime \prime }(x)(z-c),$ potom ak dosad\'{\i}me do funkcie $g$ dostaneme: $g(u)=f\,$\/$\,^{\prime \prime }\left( x\right) \left( z-c\right) (u-c)>0.$ Preto\v{z}e $\left( z-c\right) (u-c)>0$, tak $f\,$\/$\,^{\prime \prime }\left( x\right) >0.$ Body $u,\,z\in I$ boli \v{l}ubovo\v{l}n\'{e}, preto tvrdenie plat\'{\i} pre ka\v{z}d\'{e} $x$ z vn\'{u}tra intervalu $I$. $\blacksquare $ \begin{center} \begin{tabular}{|c|} \hline \hyperref{{\small Sp\"{a}\v{t}}}{}{}{M66.tex#1} \\ \hline \end{tabular} \end{center} \end{document}