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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

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


\textbf{Answer}

\begin{eqnarray*}
f_1 & = & -\sin x\\
f_2 & = & -\sin y\\
A & = & f_{11} = -\cos x\\
B & = & f_{12} = 0\\
C & = & f_{22} = -\cos y.
\end{eqnarray*}
The critical points are $(m\pi, n\pi)$ where $m$ and $n$ are integers.

Here $B^2-AC = -\cos (m\pi) \cos (n\pi) = (-1)^{m+n+1}$ which is
negative if $m+n$ is even, and positive if $m+n$ is odd.

$m+n$ even: $\Rightarrow$ $f$ has a saddle point at $(m\pi, n\pi)$.
$m+n$ odd and $m$ is odd: $\Rightarrow$ $f$ has local (and absolute)
minimum value, $-2$, at $(m\pi, n\pi)$.
$m+n$ odd and $m$ is even: $\Rightarrow$ $f$ has a local (and
absolute) maximum value, $2$, at $(m\pi, n\pi)$.


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