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I have to find the maximal volume of a Cuboid inscribed inside half of the Ellipsoid

D={(x,y,z): x^2/a^2 + y^2/b^2 + z^2/c^2 <=1, z>=0 }

So I decided to use Lagrange's multipliers.

That's what I got:

v(x,y,z) = 4xyz

d(x,y,z) = x^2/a^2 + y^2/b^2 + z^2/c^2 = 1

∇v = λ ∇d

2yz = λ x/a^2

2xz = λ y/b^2

2xy = λ z/c^2

⇒ y/x = b/a and z/y = c/b

y^2/b^2 = 1/3 ...... plug into d

y = b/√3

x = a/√3

z = c/√3

Vmax = 4/9 √3 abc

Answer: Vmax = 4/9 √3 abc

Is this OK?