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26A06-ExampleOfIncreasingdecreasingmonotoneFunction.tex
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26A06-ExampleOfIncreasingdecreasingmonotoneFunction.tex
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\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{ExampleOfIncreasingdecreasingmonotoneFunction}
\pmcreated{2013-03-22 13:36:08}
\pmmodified{2013-03-22 13:36:08}
\pmowner{Johan}{1032}
\pmmodifier{Johan}{1032}
\pmtitle{example of increasing/decreasing/monotone function}
\pmrecord{4}{34229}
\pmprivacy{1}
\pmauthor{Johan}{1032}
\pmtype{Example}
\pmcomment{trigger rebuild}
\pmclassification{msc}{26A06}
\endmetadata
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%%%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\begin{document}
The function $f(x)=e^{x}$ is strictly increasing and hence strictly monotone. Similarly $g(x)=e^{-x}$ is strictly decreasing and hence strictly monotone. Consider the function $h : [1,10] \mapsto [1,5]$ where $h(x)=\sqrt{x-4\sqrt{x-1}+3}+\sqrt{x-6\sqrt{x-1}+8}$. It is not strictly monotone since it is constant on an interval, however it is decreasing and hence monotone.
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\end{document}