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Suppose $f(x) = x^3 +x -1$.
\begin{enumerate} 
\item Explain why $f$ has a root in the interval $\,[0,1]$.

\item 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
$[0,1]$.

\item Calculate $g\left(\frac{1}{3}\right)$ and
$g\left(\frac{2}{3}\right)$. Show that if $A$ is large enough, $g$
must have three roots in the interval $[0,1]$. For what values of $A$ do we know for sure that $g$ has three roots in $[0,1]$?
\end{enumerate}

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