\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Suppose $y=f(x)$ is a function with
domain \underbar{all real numbers}. Show that the following is {\it
impossible}\/: 

\smallskip 

{\parindent=.5in\narrower\noindent The curvature $\kappa(x)$ at every
point of the graph of $y=f(x)$ is at least 1: $\kappa(x)\ge 1$ for all
$x$.\par} \smallskip

\noindent {\bf Comment/hint} Try to understand this statement
geometrically (remember, $f(x)=x^2$ flattens out towards the edges as
$x\to\pm\infty$ so it doesn't violate the impossibility assertion!),
but verify it using calculus. One way is to integrate an inequality.

\vfil\eject\end