Find the three positive numbers whose sum is $100$ and whose product is maximal.
We did this problem in class today and I'm having difficult understand a certain part of the solution.
$$\begin{align}x+y+z& =100 \\
xyz& = xy(100-x-y)
\end{align}$$
After this point I understand everything ($\nabla f)$ etc.. But it's just the line $$xyz=xy(100-x-y)$$
I don't see where he got this from. Could someone shed some light on how he knew this is the equation he needed.
Best Answer
$$x+y+z=100\implies z=100-x-y$$
which means $$xyz=xy\times z=xy(100-x-y)$$