What methods does Maximize use like Maximize[{3×2+2√2xy,x4+y4=1},{x,y}]\mathtt{Maximize}\left[\left\{3 x^2+2 \sqrt{2} x y,x^4+y^4=1\right\},\{x,y\}\right]

I used Maximize for the maximum value of 3×2+2√2xy3 x^2+2 \sqrt{2} x y when x4+y4=1x^4+y^4=1 , x>0,y>0x>0,y>0.

What methods does Maximize of Mathematica use?
Could you show me the processes with the Lagrange multiplier?

Maximize[{3 x^2 + 2 Sqrt[2] x y, x^4 + y^4 == 1}, {x, y}]

{2 Sqrt[5], {x -> Root[-4 + 5 #1^4 &, 1],
y -> (2 Sqrt[5] – 3 Root[-4 + 5 #1^4 &, 1]^2)/
(2 Sqrt[2] Root[-4 + 5 #1^4 &, 1])
}}

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3

 

There is the Lagrange method of multipliers, see e.g. this answer How can I implement the method of Lagrange multipliers to find constrained extrema?. This does not mean that behind the scene it is implemented exactly the way as in the linked anser. See also Some Notes on Internal Implementation
– Artes
Oct 11 ’15 at 23:10

  

 

For a simpler form of your result use ToRadicals, i.e., Maximize[{3 x^2 + 2 Sqrt[2] x y, x^4 + y^4 == 1}, {x, y}] // ToRadicals // Quiet
– Bob Hanlon
Oct 11 ’15 at 23:18

  

 

How to proceed with that method you can find in the linked answer. Is it really unclear therein?
– Artes
Oct 11 ’15 at 23:49

  

 

@Artes Thanks a lot. I got it
– Junho Lee
Oct 11 ’15 at 23:58

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1 Answer
1

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To your 2. question (shortly):

f = 3 x^2 + 2 Sqrt[2] x y;
g = x^4 + y^4 – 1;
L = f + \[Lambda] g;

points = NSolve[{Grad[L, {x, y}] == 0, g == 0, x > 0, y > 0}, {x,y, \[Lambda]}, Reals]
{{x -> 0.945742, y -> 0.66874, \[Lambda] -> -2.23607}}

f /. points
{4.47214}

with NMaximize

NMaximize[{3 x^2 + 2 Sqrt[2] x y, x^4 + y^4 == 1}, {x, y}]
{4.47214, {x -> 0.945742, y -> 0.66874}}