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\title{Math 241, Lecture 21}
\maketitle

\section{ Applied optimization problems}  

First, we should remember how to find the largest and/or smallest values
of a function over some interval.

\begin{itemize}
\item Find all stationary points   (by finding the derivative and setting it to zero).  
\item Find all singular points.  
\item Find the endpoints   (these are often given).  
\item Compared values at all of these points - the largest is the max and the smallest
is the min.  
\end{itemize}

In applied problems one usually 
has to work in order to find a single function to maximize.  
One needs to keep track of relationships and constraints.  

\begin{example}
Suppose advertising costs \$1000 per unit (say for magazine adds), and
product development costs \$20000 per unit.     Suppose that the profits
generated from $x$ units of advertising and $y$ units of product development are
$xy^2$ thousands of dollars.     If a company has \$10000 to spend on 
advertising and product development together, how should the money be
allocated in order to maximize profits?  
\end{example}


Applied problems are not as ``cut and dried'' as other exercises, but there are some
elements common to most such problems.  

\begin{itemize}
\item Identify all variables involved.   They often need to be named.  
\item Identify which variable needs to be maximized or minimized.  
\item Identify all  relationships between and constraints on variables.  
\item (Optional) Restate problem abstractly in the form ``find the maximum of
the function ... with the constraint(s) that ...''
\item Use the relationships to express one variable as a function
of one other variable.  
\item Use our optimization techniques above to find the minimum or
maximum of the appropriate variable.  
\end{itemize}

\begin{example}
Farmer Fred has 100ft of fencing to use to enclose his sheep in a rectangular area
next to a river.   What is the largest area which he can enclose?  
\end{example}

\begin{example}
Suppose the top and bottom of a box is made of a metal which costs 10 cents per square
centimeter and the sides are made of a metal which costs 12 cents per square centimeter.  
What is the largest volume can which can be made from two dollars of material?  
\end{example}

\begin{example}
Suppose the daily production level at a factory is modeled by a Cobb-Douglas production 
function $P = L^{0.7} C^{0.3}$, where $L$ is the number of workers and $C$ 
is the cost of materials measured in thousands of dollars.     If each worker costs
the company \$200 per day, what is the maximum production level which can be
achieved with a total cost of \$10000 per day. 
\end{example}
  



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