\be
\item Express each of the following NLPs in the general form
(\ref{E:overview_NLP_general_NLP_form}). Then create a contour diagram for the objective function, and use your result to estimate the solution of the NLP. Check your estimate using
Maple's \mf{NLPSolve} command.
\be
		\item \begin{align*}
		\mbox{minimize } f(x_1,x_2)&=x_1^2-6x_1+x_2^2-4x_2  \\
		\mbox{ subject to} \notag\\
		x_1+x_2 &\leq 3  \notag\\
		x_1\phantom{+x_2} &\leq 1 \notag \\
		x_2 &\geq 0\notag
		\end{align*}	
		\item \begin{align*}
		 \mbox{minimize } f(x_1,x_2)&=x_1^3-2x_1-x_2   \\
		\mbox{ subject to} \notag\\
		x_1+x_2 &\leq 1  \notag\\
		x_1,x_2&\geq 0 \notag
		\end{align*}	
		\item \begin{align*}
		 \mbox{maximize } f(x_1,x_2)&=x_1^2x_2-x_1  \\
		\mbox{ subject to} \notag\\
		-x_1+x_2 &\geq 1  \notag\\
        x_1^2+x_2^2 &\leq 2  \notag
		\end{align*}	
		\item \begin{align*}
		 \mbox{minimize } f(x_1,x_2)&=4x_1^2-12x_1+x_2^2-6x_2 \\
		\mbox{ subject to} \notag\\
		x_1+x_2^2 &\leq 2 \notag \\
        2x_1+x_2&=1 \notag
		\end{align*}
        \item \begin{align*}
		 \mbox{maximize } f(x_1,x_2)&=\ln(x_2+1)-x_1^2\\
		\mbox{ subject to} \notag\\
		(x_1-1)^2+(x_2-2)^2 &\leq 1 \notag\\
        -2x_1+x_2&\leq 1 \notag
		\end{align*}
        \item \begin{align*}
		 \mbox{minimize } f(x_1,x_2)&=\frac{x_1+2}{x_2+1}\\
		\mbox{ subject to} \notag\\
		x_1+2x_2 &\leq 3 \notag\\
        -x_1+x_2&\leq 1 \notag
		\end{align*}
	
\ee

\item (\emph{Pam's Pentathlon Training Program}\index{Pam's Pentathlon Problem}) The women's pentathlon consists of five events:
the 800 meter run, the 60 meter hurdles, the high jump and long jump, and the shot put
throw.\footnote{Based upon Ladany, \cite{Ladany_pentathalon}, (1975).} Pam has decided to begin an
 off-season training program combining weight lifting, distance running, and speed workouts (e.g., sprints) in order to improve her performance in her weakest two of the five events: the 800 meter run and the shot put. Each week Pam plans to devote a certain number of hours to each of these three activities, with the number devoted to each remaining constant from one week to another.

\newpar
Currently, Pam completes the 800 meter run in 3 minutes and throws the shot put 6.7 meters. Pam's coach estimates that for each hour per week Pam devotes to weight lifting, she will decrease her 800 meter run time by half that many seconds and increase her shot put distance by one-tenth of a meter. Thus, for example, lifting weights two hours per week will decrease her 800 meter run time by 1 second and increase her shot put distance by .2 meters. Similarly, the coach estimates that for each hour per week Pam devotes to distance running and for each hour per week Pam devotes to speed workouts, she will decrease her 800 meter run time by that
many seconds.

\noi
Pam's workout routine must adhere to certain constraints, which are as follows:
\be
\item Pam is expected to train between six and ten hours per week.
\item Between two and four hours of this time must be devoted to weight lifting.
\item Between three and four hours should be spent distance running.
\item In order to ensure Pam devotes sufficient time to speed workouts, she must
allot at least 60\% of her total running time to this activity.
\ee

According to International Amateur Athletic Federation guidelines, the number of points contributing to Pam's total pentathlon score by her 800 meter run performance is determined
using the formula, $P_1=.11193(254-t)^{1.88}$, where $t$ is the time, in seconds, required
for her to complete
the run. Similarly, the number of points contributing to Pam's total pentathlon score by her shot put performance equals
 $P_2=56.0211 (d-1.5)^{1.05}$, where $d$ is the distance, in meters, of her throw.

\newpar
Based upon this information, set up and solve an NLP having three decision variables and ten constraints (including sign restrictions), whose solution
indicates the number of hours Pam should devote weekly to weight lifting, distance running, and speed workouts in order to maximize her score in the 800 meter run and shot put components of the pentathlon. By how much will her total score increase?


\ee

