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 April 29th, 2017, 07:52 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, 04:57 PM #2 Senior Member   Joined: Sep 2015 From: USA Posts: 2,430 Thanks: 1315 $f(x,y)=y \sin(x)$ $f_x=y\cos(x)$ $f_y=\sin(x)$ 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 Tags extrema, local Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post life24 Calculus 7 May 16th, 2016 03:34 PM Mrto Calculus 3 April 23rd, 2016 03:26 PM mike1127 Calculus 2 March 22nd, 2016 12:17 AM CPAspire Pre-Calculus 2 March 27th, 2015 07:52 AM crnogorac Calculus 1 December 24th, 2013 04:03 AM

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