I am trying to evaluate the $\lambda$ in the result of the maximization problem of a box with a cardboard sheet of area $c$. The model is:
$$\begin{align} \text{maximize} \quad &f(x,y,z)=xyz, \\ \text{subject to} \quad &2xy+2xz+2yz=c. \end{align}$$
The solution is $x=y=z=\frac{\sqrt{6c}}{6}$ and $\lambda=-\frac{\sqrt{6c}}{12}$. Thus, $f(x,y,z)=(\frac{\sqrt{6c}}{6})^3=\frac{\sqrt{6}c^\frac{3}{2}}{36}$, and $\frac{\partial f(x,y,z)}{\partial c}=\frac{\sqrt{6c}}{24}$.
As you can see $\lambda\neq\frac{\partial f(x,y,z)}{\partial c}$, but $\lambda=-2\cdot \frac{\partial f(x,y,z)}{\partial c}$. Why is this happening if the theory states that the shadow price of a constraint is $\frac{\partial f(x,y,z)}{\partial c}=\lambda$?