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October 21st, 2013, 06:21 AM  #1 
Newbie Joined: Aug 2013 Posts: 1 Thanks: 0  Find the absolute minimum and absolute maximum.
Find the absolute minimum and absolute maximum of f(x,y) = x^2+y^22(xy) in the region defined by the x^2+y^2?4 ? 
October 21st, 2013, 07:03 AM  #2 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: Find the absolute minimum and absolute maximum.
When you're looking for minima and maxima, the first thing to try is to differentiate. In this case, as f is afunction of x and y, you need to take partial derivatives wrt x and y. So, the partial derivatives of f are 0 when x = 1 and y = 1. And, as both second derivatives are positive, these represent local minima, hence an absolute minimum for f at (1, 1). The maxima must then occur on the boundary, as there are no local maxima for f. On the boundary, we have: You can you can use more calculus to show that a maximum occurs at: 
October 21st, 2013, 06:53 PM  #3  
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Find the absolute minimum and absolute maximum. Quote:
We could replace y with . That would be the hard way! We could switch to polar coordinates with r= 2 (since is a circle with center at (0, 0) and radius 2) to have and so we are finding min and max of for . That's fairly straight forward. We could use "Lagrange Multipliers" to minimize f(x,y)= 4 2x+ 2y subject to the condition that . and . We must have for some number so we must have and , together with , three equations to solve for the three unknowns, x, y, and . Since a specific value of is not necessary for a solution, I find it is often best to eliminate first by dividing one equation by another. Here, that gives which reduces to so y= x. Together with , we have so that . Together with y= x, we have two solutions, and .  
October 22nd, 2013, 12:38 AM  #4 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: Find the absolute minimum and absolute maximum.
It might be useful to remember that on a circle x+y has a maximum when . In an exam, you'll have to work this out every time(and using polar coordinates is the best way to go), but it's the sort of thing I just remember. In this case we had to maximise yx, so that would be at While is the minimum on the boundary. Also: 
October 22nd, 2013, 01:29 AM  #5 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  Re: Find the absolute minimum and absolute maximum.
When they say 'absolute minimum' are they referring to 'global minimum' or 'minimizer of the absolute value'? I believe they're referring to the second.

October 22nd, 2013, 01:31 AM  #6 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  Re: Find the absolute minimum and absolute maximum.
My bad, they're referring to global extrema, which is the first. I just checked.


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