\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Suppose $n$ is a positive integer,
and $f$ is the function $f(x)=nx^{(n^2)}$. For example, if $n=5$,
$f(x)=5x^{25}$.

\medskip

\noindent a) What is the \underbar{largest value} of $f$ on the unit
interval, $[0,1]$? Your answer will depend on $n$. What happens to
this value as $n\to \infty$?

\medskip

\noindent b) What is the \underbar{average value} of $f$ on the unit
interval, $[0,1]$? Your answer will depend on $n$. What happens to
this value as $n\to \infty$?

\medskip

\noindent c) The asymptotic behavior of the answers to a) and b) are
different as $n\to \infty$. Briefly explain why this is possible. You
may refer to graphs of functions if that is helpful.

\vfil\eject\end