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: 1,974 Thanks: 1025 
$f(x,y)=y \sin(x)$ $f_x=y\cos(x)$ $f_y=\sin(x)$ 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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Local Minima and Local Maxima  life24  Calculus  7  May 16th, 2016 03:34 PM 
Local extrema  Mrto  Calculus  3  April 23rd, 2016 03:26 PM 
critical numbers and local extrema  mike1127  Calculus  2  March 22nd, 2016 12:17 AM 
Proving local extrema using variables.  CPAspire  PreCalculus  2  March 27th, 2015 07:52 AM 
Local Extrema  crnogorac  Calculus  1  December 24th, 2013 04:03 AM 