\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} A graph of the derivative of $f(x)$
is displayed below. Information about the function $f(x)$ is known
only for $-2.5<x<3.5$. Also {$\underline{f(-2)=1}$}. Consider the
graph carefully, and consider the information in both the numbers and
the shapes of the graph (both ``quantitative'' and ``qualitative''
information)!

\bigskip
\centerline{\epsfxsize=3.5in\epsfbox{w3R.eps}}

\centerline{Graph of $y=f'(x)$, the derivative of $f(x)$}

\medskip

\noindent a) Explain why $-2<f(0)<-1$. Look carefully at the
graph and make estimates using the MVT. Explain the steps of your
reasoning in detail.

\medskip

\noindent b) Explain why $f(3)>4+f(1)$. Again, use the MVT and explain
your reasoning in detail.

\smallskip

\noindent c) How big and how small can $f(1)-f(0)$ be? 

\smallskip

\noindent d) Use the information in a), b), and c) to explain why
$f(3)$ must be positive.

\smallskip

\noindent e) Explain why $f(x)=0$ must have a solution between $0$ and
$3$. Use the IVT and the information obtained in previous parts of
this problem.

\smallskip



\vfil\eject\end