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  pdfauthor={Dev Sinha, University of Oregon},
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\begin{document}

{\Large Stationary and singular points} \pz

\begin{definition}
A number $x$ in the domain of a function $f(x)$ is a stationary  number if $f'(x) = 0$ \pz
and is a singular number if  the derivative at $x$ does not exist.  \pz 
\end{definition}

\begin{example}
Find all the stationary points for the function $f(x) = x^3 - \frac{9}{2}x^2  + 2x -5$. \pz
Find all the singular points of the function $g(x) = | 5 +4x - x^2 |$. \pz
Graph these functions and say what you see at critical points. \pb
\end{example}

And we mentioned last time that finding stationary and singular points was the key
to understanding where a function had relatively high or low values. \pz

\begin{example}
Find where the following functions are increasing or decreasing, \pz then find
the stationary points and graph the function near those stationary points. \pz
\begin{itemize}
\item $-\frac{1}{3}x^3  + 4x - 27.$ \pz
\item $e^x - x$. \pz
\item $\frac{e^x}{x - 2}$ \pz
\end{itemize}
\end{example}

{\Large Relative extrema} \pz

We see that if a function changes from increasing to decreasing at a stationary or singular point,
that function obtains some kind of maximum \pz (there is a similar statement of
minima, which you should think about). \pz We now formalize this. \pb

\begin{definition}
Let $P = (c, f(c))$ be a critical point of a function $f(x)$. \pz
\begin{itemize}
\item $P$ is a relative maximum if $f'(x) > 0$ for nearby $x<c$ and $f'(x) < 0$ for 
nearby $x> c$. \pz
\item $P$ is a relative minimum if $f'(x) < 0$ for nearby $x<c$ and $f'(x) > 0$ for nearby
$x> c$. \pz
\item $P$ is not a relative extremum if $f'(x)$ has the same sign on both sides of
$c$. \pb
\end{itemize}
\end{definition}

\begin{example}
Find the relative maxima and minima of the functions from the previous example. \pz
\end{example}

If we are looking for largest or smallest values of a function, it often depends on
where the function is defined.  \pz In particular, a relative maximum or minimum can
also occur on an endpoint of the domain. \pz  For example, you were shortest at the
moment you were born, so time $t=0$ is where your height function has a minimum. \pb

We now include endpoints of the domain whenever we refer to critical points \pz 
(because it is critical to check for maxima and minima at those points!). \pz

\begin{definition}
If $f$ is defined only over an interval from $a$ to $b$, we say that $f$ has a relative
minimum at $a$ if $f'(x)$ is positive for values of $x$ close to $a$.  \pz We say that ...
(you fill in the rest!). \pz
\end{definition}

\begin{example}
Classify the critical points (as relative maxima, relative minima, or neither) \pz
of $\frac{x^2 - 3}{e^x}$ over the interval from 
$-4$ to $4$.  \pz  Check the answer with the graph. \pz
\end{example}

\begin{example}
Classify the critical points of the function $\ln (|x^2 - 2| + 1)$ over the interval from $-e$
to $e$. \pz
\end{example}

\begin{example}
The revenue from Pet Rocks was 
$$R(t) = \frac{63t - t^2}{t^2 + 63},$$ 
where revenue is measured in millions of dollars and time is measured in weeks after
June 5, 1967.
When is the revenue at its maximum? \pz What is that maximum? \pb
\end{example}


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