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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

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


\textbf{Answer}

$f(x,y)=\cos (x+y)$
$$f_1=-\sin (x+y) = f_2$$
All points on the lines $x+y=n\pi$ ($n$ is an integer) are critical
points.

If $n$ is even: $f=1$ at such points.

If $n$ is odd: $f=-1$ there.

Since $-1 \le f(x,y) \le 1$ at all points in $\bf{R}^2$, $f$ must have
local and absolute maximum values at points $x+y=n\pi$ with $n$ even,
and local and absolute minimum values at such points with $n$ odd.

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