Intuition behind Lagrangian multiplier in this problem What shape should a rectangular box with a specific volume be in order to minimize its surface area?
Let the lengths of the box's edges be x, y, and z. Then the constraint of constant volume is simply g(x,y,z) = xyz - V = 0, and the function to minimize is f(x,y,z) = 2(xy+xz+yz).
Now applying lagrangian method (making differntial of the lagrangian equal to zero), (using L as the lagrangian multiplier)
we get these 4 equations,
2y + 2z - Lyz = 0
2x + 2z - Lxz = 0
2y + 2x - Lxy = 0
xyz = V
Solving these, I get 
x = y = z = 4/L
I understand that x = y = z is the minimum of our surface area minimization problem. But what does the L mean here, in terms of volume and surface area?  

The correct problem handling is as follows
First the Lagrangian
$$
L(x,y,\lambda) = 2(x y+x z+y z)+\lambda(V-x y z)
$$
Second the stationary points determination by solving
$$
\nabla L = \{L_x,L_y,L_z,L_{\lambda}\} = 0
$$
or
$$
-\lambda y z + 2 (y + z) = 0\\
-\lambda x z + 2 (x + z) = 0\\
-\lambda x y + 2 (x + y) = 0\\
 V - x y z = 0
$$
Four equations with four unknowns. After solving we have
$$
x = V^{1/3}, y = V^{1/3}, z = V^{1/3}, \lambda = 4/V^{1/3}
$$
without ambiguities. 
