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\documentclass[12pt]{article}
\usepackage{FYM3}      % Style file for tests, finals

\renewcommand{\topmargin}{0.5in} % THIS MAY NEED TO BE ADJUSTED <=====

%=======================================================================
% adjust for your own test...

\renewcommand{\instructor}{\color{red}Instructor's name} 
                                                   % Instructor's name 
                                                   %   (Put your name here!)
\renewcommand{\course}{MAT 211}                    % Course number
\renewcommand{\coursetitle}{Mathematics for Business Analysis}
                                                   % Course title
\renewcommand{\examnumber}{Test \#2}               % Test type and number
\renewcommand{\examform}{Form A}                   % test form
\renewcommand{\examsemester}{Spring}               % Semester
\renewcommand{\examyear}{2007}                     % year

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\renewcommand{\pages}{\pageref{last}}
\renewcommand{\problems}{\ref{last}}
\renewcommand{\lastproblem}{\ref{lastpr}}

\def\ds{\displaystyle}

\begin{document}

\thispagestyle{empty}

\exampage    % Sets up the cover page of the test

\begin{enumerate}

\item  (10 points) Find the global minimum and the global maximum of the function
$$ f(x,y) = 2 x^2 + 3 y^2 - 5 x + 2 y + 10 $$
on the region bounded by the $x$-axis, the $y$-axis, and the line with equation $y+5=x$.
Make the picture!


\newpage

\item  Find the maximum and the minimum of the following functions under constraints:

\begin{enumerate}

\item (10 points) $ f(x,y) = 3x - 5y $ under constraint $ 5 x^2 - 2xy + 3y^2 = 10 $

\vfil \vfil

\item (10 points) $ g(x,y) = 2 x y $ under constraint $ 2 x^2 + 3 y^2 = 20 $.

\end{enumerate}

\vfil

\newpage

\item (8 points) Find the maximal and the minimal value of the function $ f(x,y) = 5x - 3y $
subject to the inequalities: $ x \ge 0 $, $ y \ge 0 $, $ x + y \le 10 $, and $ 2x + y \le 16 $.
{\bf You must
make the picture of the region!}

\newpage

\item (12 points) A factory is producing and selling white and wheat bread. For one loaf of white bread,
they need 1 pound of flour and 2 ounces of yeast.   For one loaf of wheat
bread, they need 1.5 pounds of flour and 1 ounce of yeast. The factory has
1000 pounds of flour and 700 ounces of yeast. White bread sells for \$1 per
loaf, while wheat bread sells for \$1.30 per loaf. Determine the optimal
regime of work for the factory: how many loaves of each type of bread should
be produced, in order to bring maximal revenue.

\newpage

\item (6 points) Find the row-reduced echelon form of the following matrix.
You may use your calculator. Indicate which function you used {\bf with full syntax.}
$$     \left[\begin{array}{cccc}
                2 & 1 & 0 & -1 \\
                7 & 0 & 5 &  4 \\
                1 & 3 & -2 & 0
              \end{array} \right] $$

\vfil

\item (9 points) Given the following matrix, half-way through the process of Gaussian elimination:
$$ A = \left[\begin{array}{cccc}
                1 & 3 & 2 & 4 \\
                0 & 2 & 5 & -1 \\
                0 & -5 & 7 & 2
              \end{array} \right] $$
The next step is to transform the second column. What row operations should be applied? Describe the
operations, apply them, and show the result of each operation. If you use your calculator, indicate the
functions used with full syntax. You can't use just rref!

\vfil

\newpage

\item  Solve the following systems using the {\bf reduced row-echelon forms} of the relevant matrices. You
may use your calculator. Show your work.

\begin{enumerate}

\item (5 points) $ \cases{ x-y=15 \cr 2x - 3y = 5} $

\vfil

\item (5 points) $ \cases{ 3x + 4y - z = 6 \cr -x+y-z=4 \cr 2x-y=10} $

\end{enumerate}

\vfil

\newpage

\item  Below are row-reduced echelon forms for matrices of certain linear systems. For each matrix, tell
{\bf how many solutions} the system has. {\bf Explain.} If there are any, find
the solutions. If there are infinitely many solutions, {\bf find the general
formula and two particular solutions.}

\begin{enumerate}

\item (4 points) $ \left[\begin{array}{cccc}
                1 & 0 & 0 & 3 \\
                0 & 1 & 0 & -2 \\
                0 & 0 & 1 & 5
              \end{array} \right] $

\vfil

\item (4 points) $ \left[\begin{array}{cccc}
                1 & 0 & -2 & 0 \\
                0 & 1 & 4 & 0 \\
                0 & 0 & 0 & 1
              \end{array} \right] $

\vfil

\item (7 points) $ \left[\begin{array}{cccc}
                1 & 0 & -2 & 0 \\
                0 & 1 & 3 & 0 \\
                0 & 0 & 0 & 0
              \end{array} \right] $

\end{enumerate}

\vfil

\newpage

\item Given the matrices: \\
$ A = \left[\begin{array}{ccc}
                1 & 2 & 3 \\
                4 & 5 & 6
              \end{array} \right] $,
$ B = \left[\begin{array}{cc}
                1 & 0 \\
                3 & 4 \\
                0 & -1
              \end{array} \right] $,
$ C = \left[\begin{array}{cc}
                2 & -1 \\
                0 & 1
              \end{array} \right] $.

\begin{enumerate}

\item (4 points) Which of the following products exist: $ A \cdot B $, $ B
\cdot A $, $ A \cdot C $, $ C \cdot A $, $ B \cdot C $, $ C \cdot B $? {\bf
Explain why each product exists or does not exist.}

\vfil

\item (6 points) Calculate the products from (a) that exist. You may use your calculator.
However, {\bf in each product you have to calculate at least one element
manually.}

\vfil

 \label{lastpr}
% must be BEFORE THE FIRST "/end{enumerate}" - counting problems' parts

\end{enumerate}

\label{last}
% must be BEFORE THE LAST "/end{enumerate}" - counting problems and pages

\end{enumerate}

\end{document}