April 29th, 2017, 08:52 AM  #1 
Member Joined: Nov 2016 From: Kansas Posts: 73 Thanks: 1  Local extrema
f(x,y)=ysin(x) what will be the critical and saddle points for the function? 
April 29th, 2017, 05:57 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,314 Thanks: 1230 
$f(x,y)=y \sin(x)$ $f_x=y\cos(x)$ $f_y=\sin(x)$ 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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extrema, local 
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