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

Question

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

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])
}}

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

score 4 · Accepted Answer · edited Feb 01 '17 at 12:29

To your 2. question (shortly):

f = 3 x^2 + 2 Sqrt[2] x y;
g = x^4 + y^4 - 1;
L = f + λ g;

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

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

1 Answers1