\input epsf
\nopagenumbers
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\noindent {\bf Problem statement}  Suppose that $F(x,y)$ is a function all of whose second partial
derivatives exist and are continuous.  Suppose you also know:
 
\smallskip

\centerline{$\displaystyle F(0,0)=a,\ {{\partial F}\over{\partial
x}}(0,0)=b,\ {{\partial F}\over{\partial y}}(0,0)=c,\ {{\partial^2
F}\over{\partial x^2}}(0,0)=d, \ {{\partial^2 F}\over{\partial
x\partial y}}(0,0)=e,\ {{\partial^2 F}\over{\partial y^2}}(0,0)=f$}

\smallskip
 
\noindent and that $G(s,t) = F(3s+2t, st)$.
 
\medskip\noindent

\noindent a) Compute $\displaystyle{{\partial G}\over {\partial s}}$
(your answer should be expressed in terms of $s$, $t$, and partial
derivatives of $F$).

\medskip

\noindent b) Use your answer to a) to compute 
$\displaystyle\left( {\partial\over\partial t}\left({{\partial G}\over {\partial s}}\right)\right)(0,0)$
in terms of  $a$, $b$, $c$, $d$, $e$, and $f$.

\medskip

\noindent c) Compute $\displaystyle{{\partial G}\over {\partial t}}$.
 
\medskip

\noindent d) Use your answer to c) to compute $\displaystyle\left(
{\partial\over\partial s}\left({{\partial G}\over {\partial
t}}\right)\right)(0,0)$ in terms of $a$, $b$, $c$, $d$, $e$, and $f$.

\noindent
Do your answers to b) and d) satisfy Clairaut's Theorem?

\vfil\eject\end