\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} a) What is the maximum value of the
function $f(x,y)=3x+5y$ subject to the constraint $x^2+y^2=1$, and
where is it attained? Draw a picture of the constraint and the
appropriate level set of the objective function.

\medskip

\noindent b) Suppose $n$ is a positive real number. What is the
maximum value of the function $f(x,y)=3x+5y$ subject to the constraint
$x^n+y^n=1$ and where is it attained? Your answers should all be
functions of $n$.

\medskip

\noindent c) What happens to the maximum value found in b) when $n\to
\infty$?  Try to draw a picture of the constraint and the level set
when $n$ is large.

\medskip

\noindent d) What happens to the maximum value found in b) when $n\to
0^+$?  Try to draw a picture of the constraint and the level set when
$n$ is small.
 
\medskip

\noindent {\bf Comment} Graphing programs don't seem to handle the
extreme situations described in c) and d) very well. Some thought may
be necessary to sketch the situations.

\vfil\eject\end