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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.


\subsubsection{Maximum and Minimum Values.}

One of the primary applications of calculus is to \textit{optimization} problems,
finding how to configure a situation to result in the largest, or smallest,
value of a particular output. Such things as finding the price of a product
you are selling, to maximize the profit, or how to design a box to minimize
the materials required to build it, are standard examples of optimization.

From a theoretical point of view, this theorem shows that an optimization problem
will always have a solution under the right circumstances. While this is enough
for a mathematician, it does not give any help to find the point where the optimum
value occurs.

\begin{thm}
If \( f \) is continuous on a closed interval \( [a,b] \), then there is a
point \( x_{0}\in [a,b] \) where \( f \) is largest (\( f(x)\leq f(x_{0}) \)
for all \( x\in [a,b] \)), and a point \( x_{1}\in [a,b] \)where \( f \)
is  smallest (\( f(x)\geq f(x_{0}) \) for all \( x\in [a,b] \)). 
\end{thm}
The importance of this result, which we actually don't prove, is that it asserts
the existance of such a point. To be able to say that an object \emph{exists},
with whatever property we're talking about, is very powerful in mathematics.
This may seem totally obvious, but it actually isn't. In order to see that this
is non-trivial, consider the following function:

\vspace{0.3cm}
{\par\centering \includegraphics{no-max.ps} \par}
\vspace{0.3cm}

Because the function is not continuous, it can happen (and does) that there
is \textbf{no} point where the function has a maximum. The point where that
jump occurs has a value (indicated by the dark circle) that is well below the
values near that {}``peak{}'', but the top point is not the value of the function;
the function never quite gets that large, although it can get arbitrarily close
to it. 

This behavior is ruled out for continuous functions, and is one fact that makes
continuous functions {}``nice{}''. 

\begin{defn}
If \( f(x) \) is defined on an interval \( [a,b] \), a point \( x_{0} \)
where \( f(x) \) has its largest value over \( [a,b] \)is called an \textit{absolute
maximum point}, and a point \( x_{1} \) where \( f(x) \) has its smallest
value over \( [a,b] \)is called an \textit{absolute minimum point}. These are
sometimes called \textit{extreme points}, or \textit{extrema}. The values \( f(x_{0}) \)
and \( f(x_{1}) \) are the \textit{maximum value} of \( f(x) \) and the \textit{minimum
value} of \( f(x) \), and they are referred to collectively as the \textit{extreme
values} of \( f(x) \).
\end{defn}
%

\begin{defn}
A point \( x_{0} \) where \( f(x) \) has its largest value over \textit{some}
open interval \( I \) containing \( x_{0} \) is called a \textit{local maximum
point}, and a point \( x_{1} \) where \( f(x) \) has its smallest value over
\textit{some} open interval \( I \) containing \( x_{1} \) is called a \textit{local
minimum point}. The values \( f(x_{0}) \) and \( f(x_{1}) \) are a \textit{local
maximum value} of \( f(x) \) and a \textit{local minimum value} of \( f(x) \),
and they are referred to collectively as the \textit{local extreme values} of
\( f(x) \). the points where a local maximum or local minimum happend are called
\emph{local extreme points}.
\end{defn}
Often the word {}``local{}'' is left out, even when that is really what is
meant. Other authors and professors will call these {}``relative{}'' extrema. 


\subsection*{Finding extreme points.}

Where, at what point, does a maximum or minimum of \( f(x) \) occur? That is
not, in general, an easy question to answer. But, there is a way to narrow down
the search. Certainly the extreme points could be at the endpoints of the interval
\( [a,b] \). So we should always look at \( f(a) \) and \( f(b) \). But where
else? The answer is obvious if we graph the function; the maximum occurs at
the top of a hill. It occurs at the height where no part of the graph goes above
that height. This is not a big revelation, but, if the function is differentiable
at that maximum point, since that maximum-height line is never crossed, the
tangent line can't cross over the line either, so the tangent line is horizontal.
This observation is what is sometimes called \textit{Fermat's Theorem.} It has
nothing to do with Fermat's Last Theorem; unlike that one, he may very well
have proved this one.

\begin{thm}
If \( f(x) \) has a local extremum at \( c \), and if \( f'(c) \) exits,
then \( f'(c)=0 \).
\end{thm}
\begin{rem}
\( f'(c) \) may not exist at all. The function \( |x| \) has a minimum over
all \( x\in [-1,1] \) at the point 0, even though there is no derivative there.
You have to be careful with this statement. It doesn't say that any point \( x \)
with \( f'(x)=0 \) is a maximum or a minimum. It could be something like \( f(x)=x^{3} \)
at \( x=0 \), which is not an extreme point. But, if it is an extreme point,
the derivative will be 0 if it exists. We then only have to look at all the
places where \( f'(x)=0 \), or where \( f'(x) \) doesn't exist (kinks in the
graph), and the endpoints, to find where the function has its maximum or its
minimum.
\end{rem}
\textbf{\emph{Proof.}}

\begin{defn}
A point where:
\end{defn}
\begin{enumerate}
\item \( f'(x)=0 \),
\item \( f'(x) \) does not exist, or 
\item Any endpoint of the domain of \( f(x) \),
\end{enumerate}
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}.


\subsubsection{Mean Value Theorem }

Some of the results we spend a lot of time proving may seem trivial. If so,
that is because they make explicit some standard assumptions you make about
how functions behave. But functions only really behave that way if they are
sufficiently nice, and most functions (in a very real sense, most of them) are
really badly behaved. Theoretically, these results are not really difficult,
but they are serious enough to be proved carefully. 

This first theorem is always called Rolle's theorem, named after http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Rolle.html
Michel Rolle who stated and proved this result in 1691.\footnote{%
I have been teaching calculus roughly since then, and have only now learned
the guy's first name, thanks to the internet. 
} 

\begin{thm}
\textbf{\textup{{[}Rolle's Theorem{]}.}} If \( f \) is continuous on a closed
interval \( [a,b] \), and differentiable on the open interval \( (a,b) \),
and if \( f(a)=f(b) \), then there is some point \( c\in (a,b) \) so that
\( f'(c)=0 \). 
\end{thm}
\textbf{\emph{Proof.}}

The only purpose that theorem has in life is to prove this next theorem, which
we will use to verify a lot of facts you would have assumed, but which need
proof. 

\begin{thm}
\textbf{\textup{{[}Mean Value Theorem (MVT){]}}} If \( f \) is continuous on
a closed interval \( [a,b] \), and differentiable on the open interval \( (a,b) \),
then there is some point \( c\in (a,b) \) so that \[
f'(c)=\frac{f(b)-f(a)}{b-a}.\]

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

Now, the Mean Value Thorem is often illustrated with the sort of example that
shows the statement, but not the meaning, of the theorem. In a sense, what the
statement of the theorem is, is that the slope of the curve at some point is
the same as \( (f(b)-f(a))/(b-a) \). 

\vspace{0.3cm}
{\par\centering \includegraphics{mvt.ps} \par}
\vspace{0.3cm}

But what the theorem really says is that there exists a point \( c \) with
a particular value for the derivative. That is a different viewpoint. The corollary
results in the next section generally make use of this viewpoint.

Here are two examples, one illustrating an unfortunately realistic application
of the statement, but not the idea, and the other really getting at the idea
of the MVT.

\begin{example}
My commute to work involves a stretch of the Northeast Extension of the Pennsylvania
Turnpike. I get on (at Plymouth Meeting), mile marker 20, and am handed a ticket,
with the time stamped on it. When I get off, at the Quakertown exit, mile 44,
21 minutes later, the ticket taker hands me my change and a speeding citation.
Have I been wronged?
\end{example}

\subsubsection*{Solution}

True enough, this is an application of the MVT. But the real import of this
theorem is that it provides a statement about the existence of certain values
of the derivative. Here is a better example.

\begin{example}
Show that the function \( f(x)=x^{5}+4x^{3}+3x-8 \) has only one root, one
point \( x \) where \( f(x)=0 \).
\end{example}

\subsubsection*{Solution}

\begin{xca}
Show that the function \( f(x)=x^{6}+5x^{4}+2x^{2}-7x+3 \) has at most two
distinct roots.
\end{xca}

\subsubsection*{Answer}

\textbf{\emph{Hint:}} Look at the roots of \( f'(x). \) 
\end{document}
