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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}

The presentation here, as in the textbook, jumps into a different mode. The
idea here is not just to describe, measure, and compute derivatives, but to
\emph{use} those ideas and derive consequences of them. This is a brief sojourn
into theoretical mathematics.

is called a \textit{critical point} of \( f(x) \). The values of \( f(x) \)
at a critical point are called \textit{critical values}, or \textit{critical
numbers}.

\begin{example}
Find the local maximum and minimum points and values, and the absolute maximum
minimum points and values, of the function \[
f(x)=2\sin (x)+\cos (2x),\: x\in [0,2\pi ].\]

\end{example}

\subsubsection*{Solution}

This example illustrates a basic observation about critical points, usually
called the first derivative test, since it tests whether a critical point is
a local maximum or minimum by examining the first derivatives. From Corollary
3 of the href=\char`\"{}mvt-corollaries.html\char`\"{} next section, the proof
of this result is easy.

\begin{thm}
\textbf{\emph{{[}First Derivative Test{]}}}. If \( x_{0} \) is a (non-endpoint)
critical point of a function \( f(x) \), and:

\textbf{1)} \( f'(x)>0 \) for \( x<x_{0} \) (but \( x \) near \( x_{0} \)),
and \( f'(x)<0 \) for \( x>x_{0} \), then \( x_{0} \) is a local maximum
point of \( f(x), \) 

\textbf{1)} \( f'(x)<0 \) for \( x<x_{0} \) (but \( x \) near \( x_{0} \)),
and \( f'(x)>0 \) for \( x>x_{0} \), then \( x_{0} \) is a local maximum
point of \( f(x) \).
\end{thm}
I think of this as {}``increasing up to a maximum, decreasing beyond it{}'',
or {}``decreasing down to a minimum, and increasing away{}''. It really is
useful to find the signs of the derivative and lay them out on a number line
as in the previous example, 

\vspace{0.3cm}
{\par\centering \includegraphics{derivative-signs.ps} \par}
\vspace{0.3cm}

since then you can immediately tell which are local maxima, and which are minima. 

There is another derivative test, called the second derivative test. It's not
quite so good as this one, though, for two reasons. The first is that you have
to assume that \( f'(x_{0}) \) exists (note that the first derivative test
does not mention that), and the second is that there is an indeterminate case. 

\begin{thm}
\textbf{\emph{{[}Second Derivative Test{]}.}} If \( f'(x_{0})=0, \) and 

1) \( f''(x_{0})>0 \), then \( f(x) \) has a local minimum at \( x_{0} \), 

\textbf{2)} \( f''(x_{0})<0 \), then \( f(x) \) has a local maximum at \( x_{0} \).\\
But, if \( f''(x_{0})=0 \), it could be either a local maximum, or maybe a
local minimum, or maybe neither one.
\end{thm}
\begin{xca}
Find the local and absolute maximum and minimum points, and their values, of
the function \[
f(x)=x^{4}-2x^{3}+x^{2}+1,\]
for \( x\in [-1,2] \).
\end{xca}
\textbf{Answer:}
\end{document}
