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

  1. What methods does Maximize of Mathematica use?

  2. 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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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}}
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