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\begin{document}

\textbf{ECON 4113. HOMEWORK 1 (100 POINTS). DUE THURSDAY JANUARY 26 IN CLASS.}

For all of the following problems I would like to see your work, not just answers. You should be able to solve everything below if you read Simon and Blume. For your reference: very useful web-sites for finding definitions of mathematical terms:

http://planetmath.org/

http://mathworld.wolfram.com/
\begin{enumerate}
\item (15 points) Consider a function $(x,y,z) \mapsto (x^2+2y^2+3z^2)e^{-(x^2+y^2+z^2)}$ where $x,y,z\in \Real$ (here is another useful piece of notation: people use the symbol $\mapsto$ when they don't want to give the function a name). Find all critical points (those where Jacobian derivative is $0$).
\item (10 points)Find the determinant and the rank of the following matrix:
\begin{displaymath}
\left( \begin{array}{ccc}
1&2&3\\
0&5&6\\
1&0&8
\end{array} \right)
\end{displaymath}
\item (15 points) Find out if the following matrix is positive definite, negative definite or indefinite:
\begin{displaymath}
\left( \begin{array}{ccc}
1&2&0\\
2&4&5\\
0&5&6
\end{array} \right)
\end{displaymath}
\item (25 points) Let $F:\Real^3\to\Real$ be given by $F(x,y,z)=xye^{-z}$. Find the Jacobian derivative $J$ of $F$. If you think of $J$ as a function, what are its domain and target spaces?
\item (35 points) Find the Jacobian derivative $H$ of $J$ from the previous exercise. If you think of $H$ as a function, what are its domain and target spaces? How are $H$ and $F$ related? Find all critical points of $F$ and determine which of them are local maxima, local minima and saddle points.
\end{enumerate}
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