%Lagrange Multipliers
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{\Large
Lagrange Multipliers\footnote{Copyright \copyright 2002 Steve Chapin.
All rights reserved. Please address comments to young@math.ohiou.edu.}}
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\begin{enumerate}
    \item To find the points on the ellipse $4x^2 + 9y^2 = 36$ 
    that are nearest to and farthest from the point (1,1), using the
    method of Lagrange multipliers, one needs to solve the system of
    equations\\
	$2(x - 1) - 8\lambda x = 0$\\
	$2(y - 1) - 18\lambda y = 0$\\
	$4x^2 + 9y^2 - 36 = 0$\\

Carefully {\bf derive} this system by hand.  Do 
    {\bf NOT} try to solve the system by hand.
Instead, solve the system using the commands:
\begin{itemize}
\item \verb&syms L x y & \dotfill 
               (Note that we use ``L" instead of ``$\lambda$".) 
\item \verb&[L,x,y]=solve(2*(x-1)-8*L*x,2*(y-1)-18*L*y,4*x^2+9*y^2-36)&
\item \verb&double([L,x,y])& \dotfill
       (Elements in square brackets must be in alphabetical order.)
\end{itemize}
Explain what happened.  What is the nearest point?  What is the
farthest point?  Give solutions to four decimal places.

\item Adapt the procedure in \#1 to find the points on the ellipsoid
$$
    64x^2 + 144y^2 + 36z^2 = 576
$$
that are nearest to and farthest from the point (1, 1, 1).  Write
down the system you are solving and answer the questions 
above for this example.

\item
What are your observations about symbolic versus numerical computations
from \#1 and \#2?

\item Using complete sentences and standard mathematical notation,
write a brief report (1 page only), showing your hand calculations and
answering all the questions.
\end{enumerate}

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\noindent
{\sf
The system of equations resulting
from relatively straightforward Lagrange multiplier
problems can be very difficult, if not impossible, to solve in closed
form.  In this exercise \textsc{Matlab} is used to solve such systems.  Students
are asked to compare symbolic versus numerical solutions.}

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