September 16th, 2012, 09:20 AM  #1 
Newbie Joined: Sep 2012 Posts: 4 Thanks: 0  critical points of a function
Find the critical points, if they exist, for f:R^2 > R f(x,y)=x^2+y^24x+4y+5 WITH THE LINK x + y = 5 what is the link can somebody help me what do i do here help help help 
September 16th, 2012, 10:17 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: critical points of a function
You have two choices that I know of...use the "link" to substitute for either x or y into f giving you a function of 1 variable which you can the differentiate and equate to zero to find the critical values, or use Lagrange multipliers with the "link" as the constraint.

September 16th, 2012, 12:01 PM  #3  
Newbie Joined: Sep 2012 Posts: 4 Thanks: 0  Re: critical points of a function Quote:
 
September 16th, 2012, 02:45 PM  #4 
Member Joined: May 2012 Posts: 78 Thanks: 0  Re: critical points of a function
Mark's second way is not very familiar to me, so we'll stick to the first method. The link is x+y=5. Substitute for x and get x=y+5. Now we can take that link and insert it to the function : . Now we can derive and get and the critical point is y=0.5. Eay to prove that it is a minimum point. According to the link we can write x=0.5+5=4.5. The critical point of the function is (4.5,0.5). 
September 16th, 2012, 04:20 PM  #5  
Newbie Joined: Sep 2012 Posts: 4 Thanks: 0  Re: critical points of a function Quote:
found exactly what u just said http://www.youtube.com/watch?v=ry9cgNx1QV8  
September 16th, 2012, 06:16 PM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: critical points of a function
Using Lagrange multipliers, we define: subject to the constraint: Next we use the system: in our case becomes: which implies: Substitute into the constraint: Hence, we have the critical point: 
September 16th, 2012, 07:41 PM  #7  
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: critical points of a function Quote:
 

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