May 30th, 2009, 06:10 AM  #1 
Member Joined: Jun 2008 Posts: 45 Thanks: 0  largest possible value
Hi! What is the largest possible value of when ? It seems that there is more clever way to solve the problem than using this standard method from mathematical analysis, besides I don't have to find x and y when maximum is reached. I definitely think I must use something of or , but I can't get on with mine, Mathematica and Maple results. For example, taking a=1, b=2, c=3, d=4, I get , but both Mathematica and Maple give . Okay, seems to be correct, but how can get it? 
May 30th, 2009, 10:35 AM  #2 
Senior Member Joined: Nov 2007 Posts: 258 Thanks: 0  Re: largest possible value
It's equivalent to maximizing . The two gradients will be colinear: Set : Now you have a quadratic in y^2 . Not too clean but it should work. Hope this helps! Don't forget that I disregarded the initial square root. 
June 4th, 2009, 01:59 PM  #3 
Member Joined: Jun 2008 Posts: 45 Thanks: 0  Re: largest possible value
Thank you brunojoyal very much. I will try to enter into this.

June 6th, 2009, 03:06 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 19,523 Thanks: 1748 
The expression = ?((a² + b² + c² + d² + (b² + d²  a²  c²)cos(2t) + (2ab + 2cd)sin(2t))/2), where x = sin(t) and y = cos(t). Let r = ?((b² + d²  a²  c²)² + (2ab + 2cd)²). The expression has maximum value ?((a² + b² + c² + d²)/2 + r/2). (Do you see why?) 
September 5th, 2015, 06:09 AM  #5 
Senior Member Joined: Sep 2015 From: 4th Dimension Posts: 146 Thanks: 13 Math Focus: Everything (a little bit) 
I still feel there shud be a direct method ( clever one) 

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