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\textbf{Applications of Partial Differentiation}

\textit{\textbf{Extremes}}
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\textbf{Question}

Find and classify the critical points of the function

$$ f(x,y)=\frac{x}{y}+\frac{8}{x}-y$$


\textbf{Answer}

\begin{eqnarray*}
f_1(x,y) & = & \frac{1}{y}-\frac{8}{x^2} =0 \ \ \ \rm{if} \ 8y=x^2\\
f_2(x,y) & = & -\frac{x}{y^2}-1=0 \ \ \ \rm{if} \ x=-y^2
\end{eqnarray*}
For critical points: $8y=x^2=y^4$, so $y=0$ or $y=2$.

$f(x,y)$ is not defined when $y=0$, so the only critical point is
$(-4,2)$.

At $(-4,2)$ we have
\begin{eqnarray*}
A & = & f_{11} = \frac{16}{x^3}=-\frac{1}{4}\\
B & = & f_{12} = -\frac{1}{y^2}=-\frac{1}{4}\\
C & = & f_{22} = \frac{2x}{y^3}=-1
\end{eqnarray*}

Thus $B^2-AC = \frac{1}{16}-\frac{1}{4}<0$, and $(-4,2)$ is a local maximum.

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