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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{The Mean Value Theorem}

\begin{thm}
If \( f(x) \) has a local extremum at \( c \), and if \( f'(c) \) exits,
then \( f'(c)=0 \).
\end{thm}
\begin{proof}
Assume that \( f \) has a maximum at \( c \), and assume that \( f'(c) \)
exists. Could \( f'(c) \) be positive? No, because if so, since\[
f'(c)=\lim _{h\rightarrow 0}\frac{f(c+h)-f(c)}{h},\]
if \( f'(c)>0 \), then for close enough \( h \), with \( h>0 \), \[
\frac{f(c+h)-f(c)}{h}>0,\]
 which says that, in particular, \( f(c+h)>f(c) \). So, since \( f(c) \) is
the maximum, this couldn't be. Similarly, if \( f'(c)<0 \), then (take \( h<0 \)
this time) again \( f(c) \) could not have been the largest. \\
Since both of these cases are impossible, the only remaining case, \( f'(c)=0 \),
must be true. \end{proof}

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