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April 29th, 2017, 07:52 AM   #1
ZMD
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Local extrema

f(x,y)=ysin(x)

what will be the critical and saddle points for the function?
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April 29th, 2017, 04:57 PM   #2
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$f(x,y)=y \sin(x)$

$f_x=y\cos(x)$
$f_y=\sin(x)$

critical points at $f_x=0 \text{ and } f_y=0$

$f_x = 0 \Rightarrow y =0,\text{ or }x=\dfrac{\pi}{2} + k \pi, ~k\in \mathbb{Z}$

$f_y = 0 \Rightarrow x = k \pi,~k \in \mathbb{Z}$

so the critical points are

$x = k \pi,~k \in \mathbb{Z},~y=0$

now we apply the 2nd derivative text

$f_{xx}=-y\sin(x)$
$f_{yy} = 0$
$f_{xy} = \cos(x)$

$H = \begin{pmatrix}f_{xx} &f_{xy} \\ f_{xy} &f_{yy}\end{pmatrix} =

\begin{pmatrix} -y\sin(x) &\cos(x) \\ \cos(x) &0\end{pmatrix}$

$D=|H| = -\cos^2(x) < 0,~\forall x$

Thus these critical points are saddle points
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