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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

$$f(x,y)=x^2+2y^2-4x+4y$$


\textbf{Answer}

\begin{eqnarray*}
f_1(x,y) &amp; = &amp; 2x-4=0 \ \ \ \rm{if} \ x=2\\
f_2(x,y) &amp; = &amp; 4y+4=0 \ \ \ \rm{if} \ y=-1
\end{eqnarray*}
Critical point is $(2,-1)$

Since $f(x,y)\to\infty$ as $x^2+y^2\to\infty$, $f$ has a local (and
absolute) minimum value at that critical point.

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