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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

$$f(x,y)=x e^{-x^3+y^3}$$


\textbf{Answer}

\begin{eqnarray*}
f_1(x,y) & = & (1-3x^3)e^{-x^3+y^3}\\
f_2(x,y) & = & 3xy^2e^{-x^3+y^3}\\
A=f_{11}(x,y) & = & 3x^2(3x^3-4)e^{-x^3+y^3}\\
B=f_{12}(x,y) & = & -3y^2(3x^3-1)e^{-x^3+y^3}\\
C=f_{22}(x,y) & = & 3xy(3y^3+2)e^{-x^3+y^3}
\end{eqnarray*}
For critical points: $3x^3=1$ and $3xy^2=0$. The only critical point
is $(3^{-1/3},0)$.

At that point we have $B=C=0$ so the second derivative test is
inconclusive.

However, note that $f(x,y)=f(x,0)e^{y^3}$, and $e^{y^3}$ has an
inflection point at $y=0$. Therefore $f(x,y)$ has neither a maximum
nor a minimum value at $(3^{-1/3},0)$, so has a saddle point there.

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