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. :)

  • $\begingroup$ Do they come since $\frac{\delta }{\delta x} xyz = yz $ (and same for the other terms)? (wild guess) $\endgroup$ – aba Nov 12 '13 at 19:10

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 $$\begin{cases} \dfrac{\partial{L}}{\partial{x}}=0, \\ \dfrac{\partial{L}}{\partial{y}}=0, \\ \dfrac{\partial{L}}{\partial{z}}=0, \\ \dfrac{\partial{L}}{\partial{\lambda}}=0. \\ \end{cases}$$


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