Lagrange multipliers for x,y,z

Question

I have this question, I have run completely blind into.

Find by Lagrange multipliers the volume V=xyz of where the largest box with sides adding up to x+y+z = k.

I have found the gradient of V:

dV/dx = yz
dV/dy = xz
dv/dz = yx

and the gradient of g for all x,y,z are 1.
Then I have the constraint saying: x+y+z=k

Now I use Lagranges multiplier:
GradientV = lambda*gradientG.
for x: yz = lambda
for y: xz = lambda
for z: yx = lambda

Now im lost of where to go, can anybody help me?

Answer 1

The equations $xy = \lambda$, $yz = \lambda$ and $zx = \lambda$ can be combined to get $$ xy = yz = yx.$$ If one of the variables is not zero you get that the other two have to be equal. And if two of the variables are nonzero all of them have to be equal. Taking into account the constraint you end up with $$ (x,y,z) \in \left\{\left(0,0,k\right),\left(0,k,0\right),\left(k,0,0\right),\left(\frac{k}{3},\frac{k}{3},\frac{k}{3}\right)\right\}.$$