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\markright{Chapter 3: Worksheet 6}
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\Bf{Chapter 3: Worksheet 6  \hfill Jack K. Cohen \hfill Colorado School of Mines}

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\Cb{Closed Interval Extremum Problem}
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\noindent \Bf{Suggested Problems}
Section 3.5: 1, 3, 5, 9, 10, 12, 15, 19.

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\begin{enumerate}
\item The text states the important \Em{Maximum} Value Property.
	\begin{enumerate}
	\item State the corresponding \Em{Minimum} Value Property and explain why it is true (assuming that the Maximum Value Property is true).
	\item State a combined theorem that covers \Em{both} the Maximum and Minimum Value Properties.
	\item Give an example function and interval where the Maximum Value Property \Em{fails} to hold because the function is discontinuous at a single point.
	\item Give an example function and interval  where the Maximum Value Property \Em{fails} to hold because the interval of continuity is not closed at one endpoint.
	\item Give an example function and interval where the Maximum Value Property \Em{does} hold \Em{despite} the fact that the function is discontinuous on the given interval.
	\item Give an example function and interval  where the Maximum Value Property  \Em{does} hold \Em{despite} the fact that the interval of continuity is not closed.
	\item Give an example where the hypotheses of the Maximum Value Property  hold and the function attains its maximum at two points on the closed interval.  Is there anything wrong with this?	
	\end{enumerate}

\item Give examples of the following situations or explain why the stated situation is impossible.
	\begin{enumerate}
	\item $f$ attains a minimum in $[a, b]$, but does not attain a maximum there.
	\item $f$ attains neither a maximum nor a minimum in $[a, b]$.
	\item $f$ is continuous on $[a, b]$, but attains neither a maximum nor a minimum there.
	\item $f$ attains a maximum in $[a, b]$ at several points in this interval.
	\end{enumerate}

\item (3.5.16)
	\begin{enumerate}
	\item Find the maximum and minimum values attained by $h(x) = x^2 + 4x + 7$ on the interval $[-3, 0]$.
	\item Check your answers intuitively using \Tt{Plot}.
	\item Explain why your answers are rigorously correct.
	\end{enumerate}

\item (3.5.30) Find the maximum and minimum values attained by $f(x) = |x + 1| + |x - 1|$ on the interval $[-2, 2]$---also state the $x$ values where these extrema are attained.

\item Give examples of the following:
	\begin{enumerate}
	\item $f'(0) = 0$, but f(0) is neither a local maximum nor a local minimum of $f(x)$.
	\item $f'(0) \neq 0$, but $f(0)$ is a local maximum of $f(x)$.
	\item Suppose $f'(0) = 0$.  State an intuitive condition on the values of $f'(x)$ \Em{near} $x = 0$ that guarantees that $f(0)$ is a local maximum of $f(x)$.
	\item Repeat last part for local \Em{minimum}.
	\end{enumerate}

% Logistic growth
\end{enumerate}
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