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\begin{document}

\title{Math 241, Lecture 19}
\maketitle

\section{ Stationary and singular points}  

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

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

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

\subsection{ Relative extrema}  

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

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

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

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

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

\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 $a$.    We say that ...
(you fill in the rest!).  
\end{definition}

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

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


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