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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{Corollaries of the Mean Value Theorem}

\begin{cor}
If \( f'(x)=0 \) for \textbf{all} \( x\in [a,b] \), then \( f \) is a constant. 
\end{cor}
\begin{proof}
If \( f(x) \) were not constant, there would be a point \( \xi \in (a,b) \)
where \( f(\xi )\neq f(a) \). Say, for instance, that \( f(\xi )>f(a) \).
Then, by MVT, there is a point \( c\in (a,\xi ) \), where \( f'(c)=(f(\xi )-f(a)/(\xi -a)\neq 0 \),
which contradicts the fact that \( f'(x)=0 \) for all \( x \). On the other
hand, if \( f(\xi )<f(a) \), the same argument (with the signs reversed), shows
that again a contradiction is reached. The only other case is that \( f(x) \)
is \textit{never} unequal to \( f(a) \). \end{proof}

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