\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} If $f$ is a function, then a number
$x_0$ is called a {\it fixed point}\/ of $f$ exactly when
$f(x_0)=x_0$.

\medskip

\noindent a) Find all the fixed points of the following functions to
three-place accuracy.
$$
f(x)=x^2\qquad g(x)=3e^x-2e^{-x}\qquad h(x)={2\over3}\arctan x.
$$

\noindent b) Illustrate your answers graphically. Give three graphs,
each one showing $x$, one of the functions above, and any fixed
points.
 
\medskip

\noindent c) Suppose that $f$ is a differentiable function and
$f'(x)<1$ for all $x$. Use the MVT to explain why $f$ can have no more
than one fixed point. To which of the functions in a) does this
general statement apply?


 








\vfil\eject\end

