I'm currently stuck with a Lagrange optimization problem where I just can't find the right Lagrangian function.
The problem statement is as follows:
The function $f(x,y,z) = x^2+y^4+z^6-1=0$ defines a closed surface area in $\mathbb{R}^3$. This area is symmetric to the three coordinate planes $xy=0$, $yz=0$, $zx=0$. Now place a also symmetric positioned, axis-parallel, cuboid in this area in that way, that its surface area is maximised.
Formulate a system of equations which solutions give the corner points $(x,y,z)$ of this cuboid. (No need to actually solve this system of equations!)
I know that function to maximise should just be $g(a,b,c)=2(ab+bc+ac)$ and one equality constraint is the given function $f(x,y,z)=0$.
What I'm not sure about is how I get from this resulting Lagrangian function to the eight coordinates (x,y,z) (cause of eight corners) from my cuboid.
Any guidance in the right direction would be really appriciated. I've done a few problems on Langrange optimization yet but I just dont't know where to go with this one.
EDIT: I also thought about including the three equations $xy=0$, $yz=0$, $zx=0$ as equality constraints. Is there a general way to identify which equation is relevant for a equality constraint.
Best Answer
The lagrangian can be stated as
$$ L = 8(x y + x z + y z)+\lambda(x^2+y^4+z^6-1) $$
and after solving the stationary conditions
$$ \nabla L = 0 = \cases{2 \lambda x+8 (y+z)\\ 4 \lambda y^3+8 (x+z)\\ 6 \lambda z^5+8 (x+y)\\ x^2+y^4+z^6-1} $$
we have the solution
$$ \{ x = 0.739178, y = 0.720635, z = 0.754121\} $$
with a surface of $13.0684$ units.