\input epsf
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\magnification=\magstep1

\noindent {\bf Problem statement} 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:


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\centerline{%
\vtop{%
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\quad\hfil$#$\hfil\quad&\vrule#&
\quad\hfil$#$\hfil\quad&\vrule#&
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\quad\hfil$#$\hfil\quad&\vrule#&
\quad\hfil$#$\hfil\quad&\vrule#&
\quad\hfil$#$\hfil\quad\cr
   x&&  -2&&  -1&&  0&&  1&&  2\cr
\noalign{\hrule}
f(x)&&   2&&  -1&&  2&& -1&&  2\cr}}}
\medskip

\noindent a)~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 $[-2,2].$


\smallskip

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

\medskip

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

\smallskip

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

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

\medskip

\noindent c) 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.

\smallskip

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

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

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

\item{\it iv)} How many roots does $f$ have in the interval $[0,1]$?

\smallskip

\noindent Approximate the roots of $f$ in $[0,1]$ to three decimal
places using a calculator.

\medskip

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

%contributed by Michael O'Nan







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