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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

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

\textbf{Answer}

\begin{eqnarray*}
f_1 & = & 4(x^3-y)\\
f_2 & = & 4(y^3-x)\\
A & = & f_{11} = 12x^2\\
B & = & f_{12} = -4\\
C & = & f_{22} = 12y^2.
\end{eqnarray*}
For critical points: $x^3=y$ and $y^3=x$. Thus $x^9=x$, or
$x(x^8-1)=0$, and $x=0,1$, or $-1$.

The critical points are $(0,0)$, $(1,1)$ and $(-1,-1)$.

At $(0,0)$: $B^2-AC=16-0>0$, so $(0,0)$ is a saddle point.

At $(1,1)$ and $(-1,-1)$: $B^2-AC_16-144<0$, $A>0$ so $f$ has local
maxima at these points.


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