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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

$$ f(x,y)=\frac{1}{1-x+y+x^2+y^2}$$


\textbf{Answer}

\begin{eqnarray*}
f(x,y) & = & \frac{1}{1-x+y+x^2+y^2}\\
& = & \frac{1}{ \left ( x-\frac{1}{2} \right )^2 + \left ( y +
\frac{1}{2} \right )^2 + \frac{1}{2} }
\end{eqnarray*}
Obviously $f$ has maximum value $2$ at $(\frac{1}{2}, -\frac{1}{2})$.

Since
\begin{eqnarray*}
f_1(x,y) & = & \frac{1-2x}{(1-x+y+x^2+y^2)^2}\\
f_2(x,y) & = & -\frac{1+2y}{(a-x+y+x^2+y^2)^2}
\end{eqnarray*}
$\displaystyle \left ( \frac{1}{2}, -\frac{1}{2} \right )$ is the only
critical point of $f$.


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