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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

$$f(x,y)= x\sin y$$


\textbf{Answer}

For critical points we have:
$$f_1=\sin y =0 \ \ \ \ \ f_2=x\cos y=0.$$
Since $\sin y$ and $\cos y$ cannot vanish at the same point, the only
critical points correspond to $x=0$ and $\sin y=0$.

They are $(0,n\pi)$, for all integers $n$. All are saddle points.

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