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{\bf Exam Question

Topic: CriticalPoints}

Find and classify the critical points of the function
$$f(x,y)=3x^4+12xy+4y^3.$$  \vspace{0.5in}

{\bf Solution} $$f(x,y)=3x^4+12xy+4y^3.$$
$f_x=12x^3+12y=0\Rightarrow y=-x^3.$

$f_y=12x+12y^2=0\Rightarrow x=-y^2.$

So $x=-x^6,$ giving $x=0$ or $x^5=-1$ i.e. $x=-1.$

The critical points are therefore $(0,0);\ (-1,1)$ \bigskip

\begin{center}
\begin{tabular}{|c|c|c|}
\hline &$(0,0)$&$(-1,1)$\\ \hline $f_{xx}=36x^2$&0&36\\ \hline
$f_{xy}=12$&12&12\\ \hline $f_{yy}=24y$&0&24\\ \hline
$\Delta=f_{xy}^2-f_{xx}f_{yy}$&144&$-720$\\ \hline
\end{tabular}
\end{center}
\bigskip
So$(0,0)$ is a saddle point, and since $f_{xx}(-1,1)>0,\ (-1,1)$
is a local minimum.

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