I've seen this puzzle on Flammable Maths new video (https://www.youtube.com/watch?v=vW4TjU4IoPY)(optional to watch). It is as follows:
"Given a+b+c+d+e...=10, what is the biggest value for abcde... given n elements?"
I've managed to do my own solution using calculus, here it is: https://docs.google.com/document/d/1sSQN_OQCtTzafJnkSPGT7XK28QdamF2XlLDLTxLFVA0/edit?usp=sharing
Thing is, I'm accidentally only accounting for positive values of a, b, c.... If I weren't, the set (500, -489, -1), for example, would be a solution to a + b + c = 10. So it's clear negative numbers allow for infinitely large products. On my solution, I didn't consciously restrict the domain for positive numbers at all, and yet only got the positive solution. I'm wondering what step did I take that restricted my domain.
I've found the same exact solution as the one in the video, in which its assumed (a, b, c, d, e...) are all positive. The only explanation I can think to explain this is that I've accidentaly totally disregarded a set of solutions, through some operation with restricted domain, of which I don't know. So, the main question here is "Where, did on my solution, I accidentally restricted the domain of my solutions?"
Both links I posted offer different ways to find the biggest value for abc*d..., and the purpose of this post isn't to find the best solution, but to understand a possible neglect I've done on my solution regarding the domain range.