I was going though this website for a general idea of Lagrange Multipliers. One of its examples was to maximize $xyz$ given $xy+yz+zx= 32$. The first few lines of the solution went as follows:
$$ $$
Let $\lambda$ be the Lagrange multiplier of this system. Then, we obtain the following equations:
$$yz= \lambda(y+z)$$
$$zx= \lambda(z+x)$$
$$xy= \lambda(x+y)$$
$$xy+yz+zx= 32$$
I think I understand how the terms $x+y$, $z+x$, and $y+z$ come:
$$ \frac{\delta}{\delta x} (xy+yz+zx)= y+z$$
Similarly we can get the other terms. What I cannot get is the terms in the L.H.S. My question is: how are we getting the terms $xy$, $zx$, and $xy$?
I am sorry if this is something too trivial, I am just a beginner in this field. :)
P.S Note that we don't need Lagrange multipliers to solve this problem. I'm just trying to figure out how to apply them. :)
If $$L=xyz+\lambda(xy+yz+zx-32)$$ denotes the Lagrange function, then the stationary points of $L$ can be found as solutions of the system $$\begin{cases} \dfrac{\partial{L}}{\partial{x}}=0, \\ \dfrac{\partial{L}}{\partial{y}}=0, \\ \dfrac{\partial{L}}{\partial{z}}=0, \\ \dfrac{\partial{L}}{\partial{\lambda}}=0. \\ \end{cases}$$