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May 30th, 2009, 06:10 AM   #1
JC
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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?
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May 30th, 2009, 10:35 AM   #2
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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.
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June 4th, 2009, 01:59 PM   #3
JC
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Re: largest possible value

Thank you brunojoyal very much. I will try to enter into this.
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June 6th, 2009, 03:06 PM   #4
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The expression = ?((a + b + c + d + (b + d - a - c)cos(2t) + (2ab + 2cd)sin(2t))/2),
where x = sin(t) and y = cos(t).

Let r = ?((b + d - a - c) + (2ab + 2cd)).
The expression has maximum value ?((a + b + c + d)/2 + r/2). (Do you see why?)
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September 5th, 2015, 06:09 AM   #5
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I still feel there shud be a direct method ( clever one)
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