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\textbf{Applications of Partial Differentiation}

\textit{\textbf{Extremes within restricted domains}}
\end{center}

\textbf{Question}

Find the maximum and minimum values of

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

On the closed triangle bounded by $x=0$, $y=0$ and $x+y=2\pi$.


\textbf{Answer}

$$-1 \le f(x,y) = \sin x \cos y \le 1, \ \ \ \textrm{everywhere}$$
And
\begin{eqnarray*}
f(\pi/2, 0) &amp; = &amp; 1\\
f(3\pi/2,0) &amp; = &amp; -1
\end{eqnarray*}
Both $(\pi/2,0)$ and $(3\pi/2,0)$ are on the triangle.
\begin{eqnarray*}
\Rightarrow \textrm{min}(f) &amp; = &amp; -1\\
\textrm{max}(f) &amp; = &amp; 1
\end{eqnarray*}

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