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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}

There are always some problems in the text where you are supposed to find the
\( c \) which ``fits'' the statement of the MVT for a particular function.
While that may help you remember the statement, that isn't the point. The point
is that the result can be applied \textit{theoretically} to other facts, as
in these corollaries (A ``corollary'' is a theorem that is a simple consequence
of an earlier result.): 

\begin{cor}
If \( f'(x)=0 \) for \textbf{all} \( x\in [a,b] \), then \( f \) is a constant. 
\end{cor}
\begin{rem}
Hard to imagine why we need to prove this, isn't it? However, we needed to have
essentially this result (the MVT) in order to show this seemingly trivial statement. 
\end{rem}
\textbf{\emph{Proof.}} 

\begin{cor}
If \( f'(x)=g'(x) \) for \textbf{all} \( x\in [a,b] \), then, for some constant
\( C \), \( f(x)=g(x)+C \). 
\end{cor}
\begin{rem}
This statement says that, given a function \( f(x) \), if it has an antiderivative,
a function \( F(x) \) so that \( F'(x)=f(x) \), then that antiderivative can
be only one function, except possibly for a constant term, because any two antiderivatives
of \( f(x) \) have \( f(x) \) as their derivative, so differ only by a constant.
That is, there is only one way to construct an antiderivative, except for a
constant being added in, or, looking at it from \( F(x) \)'s perspective, \( F(x) \)
is completely determined by its derivative, except for that constant.
\end{rem}
\textbf{\emph{Proof.}} 

\begin{cor}
If \( f \) is continuous on \( [a,b] \), and has a positive derivative (\( f'(x)>0 \)),
on \( (a,b) \), then \( f \) is increasing on the interval \( (a,b) \). If
\( f'(x)<0 \) for \textbf{all} \( x\in [a,b] \), then \( f \) is decreasing
on \( (a,b) \). 
\end{cor}
\textbf{\emph{Proof.}} 

The next result is not really a corollary of the MVT, but is an extension of
it. It is used later on in the proof of l'H\^{o}pital's rule.

\begin{thm}
\textbf{\emph{{[}Generalized MVT{]}}}. If \( f(x) \) and \( g(x) \) are continuous
on a closed interval \( [a,b] \), differentiable on the open interval \( (a,b) \),
and if \( g'(x)\neq 0 \) for every \( x\in (a,b) \), then there is some point
\( c\in (a,b) \) so that \[
\frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}.\]

\end{thm}
\textbf{\emph{Proof.}}
\end{document}
