\documentclass{article}

\begin{document}

A continuous function $f$ is defined
on the interval $[-2,2]$.  The values of $f$ at some of the points
of the interval are given by the following table:

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}\hline
$x$ & -2 & -1 & 0 & 1 & 2 \\\hline
$f(x)$ & 2 & -1 & 2 & -1 & 2 \\\hline
\end{tabular}
\end{center}

\begin{enumerate}
\item Using only this information, what can be concluded about
the roots of $f$, that is, the solutions of $f(x) = 0$, in the
interval $[-2,2]$? The answer should be something like: $f$ has at
least 8 roots in $[-2,2]$, or $f$ has at most 6 roots in $[-2,2].$

{\bf Hint:} Use the Intermediate Value Theorem on each
of the intervals $[-2,-1]$, $[-1,0]$, $[0,1]$, and $[1,2]$.

\item If $f(x) = x^4 -4x^2 + 2$, verify that the relevant
values of $f$ are given by the table above.

\begin{enumerate}
\item Sketch the graph of $y = f(x)$ in the viewing window
$[-2.5,2.5]{\times}[-3,3]$. 

\item How many roots does $f$ have in the interval $[-2,2]$?
Find the roots algebraically. {\bf Suggestion:} Let $t = x^2$ and
solve with the quadratic formula. Then find $x$.

\end{enumerate}
\item  If $f(x) = x^4-4x^2 + 2 + 5(2x-1)x(x^2-1)(x^2-4)$.
Verify that the relevant values of $f$ are given by the table above.

\begin{enumerate}

\item Sketch the graph of $y = f(x)$ in the viewing window
$[-2.5,2.5]{\times}[-80,80]$. 

\item Explain why $f$ has at least one root in each of the
intervals $(-2,1)$, $(-1,0)$, $(0,1)$, and $(1,2)$.  

\item Sketch the graph of $y = f(x)$ in the viewing window
$[0,1]{\times}[-1,3]$.

\item How many roots does $f$ have in the interval $[0,1]$?
 Approximate the roots of $f$ in $[0,1]$ to three decimal
places using a calculator.
\end{enumerate}

\item Having done b) and c), was your original conclusion in
part a) correct?

\end{enumerate}

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