
\be
	\item For each LP below, graph the feasible region, and sketch at least three contours of the objective function. Use your contours to determine the nature of the solution. If the LP has a unique optimal solution, list the values of the decision variables along with the corresponding objective function value. If the LP has alternative optimal solutions, express the general solution in parametric form. You may wish to combine Maple's \mf{inequal}, \mf{contourplot}, and \mf{display} commands to check each solution.
	
		\be
       \item
        \begin{align*}
        \mbox{maximize } z &=6x_1+4x_2  \\
	\mbox{ subject to} \\
	2x_1 + 3x_2 & \leq 9 \\
	x_1 \phantom{ +20y} &\geq 4 \\
	\phantom{ +20x} x_2  &\leq 6  \\
	x_1,x_2 &\geq 0 \notag
    % Unique solution
	\end{align*}
		\item \begin{align*}
		 \mbox{maximize } z &=3x_1+2x_2  \\
		\mbox{ subject to} \notag\\
		x_1 \phantom{ +20x_2} &\leq 4 \notag\\
		x_1+3x_2 &\leq 15  \notag\\
		2x_1+x_2 &= 10\notag \\
		x_1,x_2 &\geq 0\notag
		\end{align*}
        % Unique Solution	

        \item \begin{align*}
		 \mbox{minimize } z &=-x_1+4x_2  \\
		\mbox{ subject to} \notag\\
		x_1 +3x_2 &\geq 5 \notag\\
		x_1+x_2 &\geq 4  \notag\\
		x_1-x_2 &\leq 2 \notag \\
		x_1,x_2 &\geq 0\notag
		\end{align*}
        % Unique Solution

        \item \begin{align*}
		 \mbox{maximize } z &=2x_1+6x_2  \\
		\mbox{ subject to} \notag\\
		x_1 +3x_2 &\leq 6 \notag\\
		x_1+2x_2 &\geq 5  \notag\\
		x_1,x_2 &\geq 0\notag
		\end{align*}
        % Alternate Optimal Solutions

        \item \begin{align*}
	       	 \mbox{minimize } z &=2x_1+3x_2  \\
		\mbox{ subject to} \notag\\
		3x_1 +x_2 &\geq 1 \notag\\
		x_1+x_2 &\leq 6  \notag\\
		x_2 &\geq 0\notag
        \end{align*}
        % No sign restriction on x1; unique solution though
		\ee

\item Use the graphical methods discussed in this section to solve the \emph{Foraging Herbivore Model},\index{Foraging Herbivore Model}
Exercise 4, from Section \ref{S:LP_Prototype_Standard_Form}.

\item  Consider the LP given by
        \begin{align*}
		 \mbox{maximize } z &=2x_1-x_2  \\
		\mbox{ subject to} \notag\\
		x_1 +3x_2 &\geq 8 \notag\\
		x_1+x_2 &\geq 4  \notag\\
		x_1-x_2 &\leq 2 \notag \\
        x_1,x_2 &\geq 0.\notag
        \end{align*}

    \be
    \item Sketch the feasible region for this LP, along with the contours corresponding to
    $z=50$, $z=100$, and $z=150$.
    \item If $M$ is an arbitrarily large positive real number, what is a description, in terms
    of $M$, of the
    feasible points $(x_1,x_2)$ corresponding to $z=M$.
    \item Explain why the LP is unbounded.
    \ee

\item Suppose $f$ is a linear function of $x_1$ and $x_2$, and consider the LP given by
\begin{align*}
 \mbox{maximize } z &=f(x_1,x_2)  \\
	\mbox{ subject to} \\
	x_1 \phantom{ +20y} &\geq 1 \\
	\phantom{ +20x} x_2  &\leq 1  \\
	x_1,x_2  &\geq 0.\notag
	\end{align*}

    \be
    \item Sketch the feasible region for this LP.
    \item Give an example of an objective function $f$ for which the LP has a unique optimal
    solution.
    \item Give an example of an objective function $f$ for which the LP is unbounded.
    \item Give an example of an objective function $f$ for which the LP has alternative
    optimal solutions.
    \ee

\ee 