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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 \( f \) 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}
\begin{proof}
Since \( f'(x)>0 \) on this interval, then for any \( x<y\in (a,b) \), there
is a \( c\in (x,y) \) so that \[
0<f'(c)=\frac{f(y)-f(x)}{y-x},\]
so \( f(y)-f(x)>0 \), or \( f(y)>f(x) \), that is, the function is increasing.
As usual, turn the inequalities around for the opposite direction. \end{proof}

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