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% John David Stone
% Department of Mathematics and Computer Science
% Grinnell College
% stone@cs.grinnell.edu

% created June 19, 2002
% last revised June 19, 2002

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{\large \textbf{Exercise set \#6---MAT 134}} \\ $\partial$ue March 10
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1. Find the greatest value of the function $xy(c - x - y)$ in the closed
triangular region with vertices $(0, 0)$, $(c, 0)$, and $(0, c)$.  Assume
$c > 0$.

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2. Find the absolute minimum value of
\begin{displaymath}
f(x, y) = x^2 + y^2 + \left( \frac{2A - ax - by}{c} \right)^2,
\end{displaymath}
where $A$, $a$, $b$, and $c$ are positive constants.  All values of $x$ and
$y$ are admitted.  How do you know that a minimum exists?

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3. Suppose $u$ depends on $x_1, x_2, x_3$ and the $x$'s depend on $\xi_1,
\xi_2$ as follows:
\begin{eqnarray*}
x_1 &=& a_{11}\xi_1 + a_{12}\xi_2, \\
x_2 &=& a_{21}\xi_1 + a_{22}\xi_2, \\
x_3 &=& a_{31}\xi_1 + a_{32}\xi_2,
\end{eqnarray*}
where the $a$'s are constants.  Write the formulas connecting
$\frac{\partial u}{\partial\xi_1}$ and $\frac{\partial u}{\partial\xi_2}$
with $\frac{\partial u}{\partial x_1}$, $\frac{\partial u}{\partial x_2}$,
and $\frac{\partial u}{\partial x_3}$.  Generalize for the case of $n$
$x$'s and $m$ $\xi$'s.

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4. If
\begin{displaymath}
F(x, y, z) = \left| \begin{array}{ccc}
                      f_1(x) & f_2(x) & f_3(x) \\
                      f_1(y) & f_2(y) & f_3(y) \\
                      f_1(z) & f_2(z) & f_3(z)
                    \end{array}                    \right| ,
\end{displaymath}
calculate $\frac{\partial F}{\partial x}$, $\frac{\partial F}{\partial y}$,
and $\frac{\partial F}{\partial z}$.
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