\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Suppose $f(x)$ is a differentiable
function with $f(-1)=2$ and $f(2)=-1$. The differentiable
function $g(x)$ is defined by the formula $g(x)=f(f(x))$.

\medskip

\noindent a) Compute $g(-1)$ and $g(2)$. Explain why $g(x)=0$ must
have at least one solution $A$ between $-1$ and $2$.

\medskip

\noindent b) Compute $g'(-1)$ and $g'(2)$ in terms of values of $f$
and $f'$. Verify that $g'(-1)=g'(2)$. Explain why $g''(x)=0$ must have
at least one solution $B$ between $-1$ and $2$.

\medskip

\noindent c) Suppose now that $f(x)=Cx^2+D$. Find values of $C$ and
$D$ so that $f(-1)=2$ and $f(2)=-1$. Compute $g(x)=f(f(x))$ directly
for those values of $C$ and $D$, and use algebra on the resulting
formulas for $g(x)$ and $g''(x)$ to find numbers $A$ and $B$ between
$-1$ and $2$ so that $g(A)=0$ and $g''(B)=0$. The ``abstract''
assertions of a) and b) should be verified.

\vfil\eject\end