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\noindent {\bf Problem statement} A {\it linear programming}\/ problem
is a max-min problem of a somewhat different character from those we
encounter in Math 251.  In the simplest case of such a problem we want
to maximize and/or minimize a linear function of $x$ and $y$, that is,
a function $f(x,y)=ax+by$, over a polygonal region like the region $R$
below.  (There are many real applications with problems of this sort,
and many more than two variables may be involved -- perhaps
thousands.)

\bigskip

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\bigskip

\noindent Suppose now that the function whose extrema we want to find
is $f(x,y)=x+2y$.

\medskip

\noindent a) Find all critical points of $f$ (pretty easy, yes?).
What does this result tell you about where the extreme values of $f$
in $R$ occur?

\medskip

\noindent b) Draw a region $R$ that looks something like the above
(the picture is not critical) and on the same figure sketch and label
some level curves of the function $f$, including some that cross $R$.
From your picture, explain why the maximum and minimum values of $f$
in $R$ will be taken on at vertices (corners0 of the region $R$, and
how you would determine the relevant vertices graphically.

\medskip

\noindent c) Suppose now that we consider a region $R$ (not
the one in the picture!) consisting of the points $(x,y)$ which satisfy
all of these inequalities: 

$$y\ge3-x,\quad 2y\ge x,\quad y\le 3x-1,\quad y\ge2x-6,\quad 3y\le17-x.$$

\noindent Make a {\it careful}\/ sketch of $R$, finding the coordinates
of all the vertices.  By adding some level curves of $f$, determine at
which vertices of $R$ the maximum and minimum of $f$ occur, and from
this find these extreme values.  Check your analysis by evaluating $f$
at all the vertices.


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