Lagrange Multiplier Problem, three variables

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.

I tried to solve this in Mathematica but I think my problem stems from
the fact my fourth equation (the original constraint equation) is not
an equation, but a function with no set value. I can get a fourth
equation V = xyz, but that obviously introduces yet another variable.
My gut feeling is that the values of x, y, and z that minimize the
function would be in terms of other variables.

When I tried to solve it in Mathematica instead of by hand, I get
something like this...


eqn1 = 2/x^3 == lambda y z
eqn2 = 2/y^3 == lambda x z
eqn1 = 2/z^3 == lambda x y
Solve[{eqn1, eqn2, eqn3},{x,y,z}

I of course get a LOT of solutions, all of them are in terms of
variables, not numbers. Can someone explain how I might be able to
solve this (by hand or Mathematica) or give justification to why there
aren't numerical solutions if that's the case?

.

Relevant Pages

  • Re: Lagrange Multiplier Problem, three variables
    ... the constraing that the volume of the box dimensions x,y,z is V = xyz. ... I tried to solve this in Mathematica but I think my problem stems from ... aren't numerical solutions if that's the case? ...
    (sci.math.symbolic)
  • Re: why are pdes so difficult?
    ... Mathematica program can't really deal with the ... >Does matlabdo any better? ... >most people told me that if you want numerical solutions to pde's you ...
    (sci.math)
  • Magnetic Flux intensity calculation using Mathematica
    ... Did anyone try to use Mathematica to calculate the magnetix flux intensity ... of a rectangular magnetic block at an arbitrary point xyz? ...
    (sci.physics.research)