Re: Lagrange Multiplier Problem, three variables

Dave Rusin schrieb:
We have been asked to solve the system

eqn1= 2/ x^3== \[Lambda] y z;
eqn2= 2/ y^3== \[Lambda] x z;
eqn3= 2/ z^3== \[Lambda] x y;
eqn4= x y z == v;

for x,y,z (with v a constant).

actually, JTolman wrote:
"I have to minimize the function f(x,y,z) = 1/x^2 + 1/y^2 + 1/z^2 under
the constraing that the volume of the box dimensions x,y,z is V = xyz."


In article <ebm069$6gh$1@xxxxxxxxx>, Peter Pein <petsie@xxxxxxxxxx> wrote:
But it doesn't need a single line of Mathematica-code to argue with symmetry and the singularities at x|y|z = 0, that the solution has x=y=z and therefore x^3==v and your min = 3*v^(-2/3).

Careful -- the equations are symmetric, and thus the solution SET has
a three-fold symmetry. Indeed x=y=z= v^(1/3) is a solution, but there
are others: we could change signs of any two of x, y, and z; and
there are complex solutions like
x = z, y = i z, z = (-1)^(1/6) v^(1/3)

"...that the volume of the box dimensions x,y,z is V = xyz."

Of course these additional solutons don't meet the other conditions
the OP had in mind, but the existence of these additional solutions
serves as a reminder that the full algebraic solution to a symmetric
system of equations can include more than just points with x=y=z.

dave


Please accept my apologies. I was not aware that citing the OP again is a necessity.

sincerely,
Peter
.