\documentstyle[fullpage, ps]{article}
\begin{document}
\header{18.021}{3}
\problem{1}
The first step is to find a function $z(x,y)$, so that $F(x,y,z) = x^3 + y^3 + z^3$ can be expressed as a function of the form $F(x,y,z(x,y)$.  The constraint equation allows one to obtain the function, $z = \frac{1}{1 - \frac{1}{x} - \frac{1}{y}}$  Substituting, and finding the points at which $F_x = F_y = 0$, yields the points $(1,1,-1), (1,-1,1), and (-1,1,1)$. 
\newcommand{\blob}[0]{1 - \frac{1}{x} - \frac{1}{y}}
$F_x = 3x^2 - \frac{1}{3x^2(\blob)}
F_y = 3y^2 - \frac{1}{3y^2(\blob)}$
Therefore,
$9x^4 = 9y^4 = \frac{1}{(\blob)}$
From this the given points are seen to be the stationary points, subject to the constraint equations.  It is also clear that these are saddle points as taking F(x+h,y+j,z+k) yeilds both greater and lesser values.

\problem{3}
\end{document}
 
