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\noindent {\bf Problem statement} Suppose $f(x) = x^3 +x -1$.

\medskip
\noindent a) Explain why $f$ has a root in the interval $\,[0,1]$.

\medskip

\noindent b) Suppose $A$ is a constant and $g(x) = x^3 +x -1
+Ax(x-1)(2x-1)$.  Show that $g$ has at least one root in the interval
$[0,1]$.

\medskip

\noindent c) Calculate $g\!\left({1\over 3}\right)$ and
$g\!\left({2\over 3}\right)$. If $A$ is large enough, show that $g$
must have three roots in the interval $[0,1]$.

%contributed by Michael O'Nan

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