Given a number $N \geq 5$. We have to divide it into equal parts so that the product of these parts is maximized. So using this I get $$f(x) = \left(\dfrac{N}{x}\right)^x$$
I differentiate this to get $x$ as $N/e$ as the point of maxima. However note that here $x$ is the "no of parts, and thus it has to be an integer, i.e., $x \in \mathbb Z_+$.
There are four options (more than one may be correct)
- $\left\lfloor \dfrac{N}{e} \right\rfloor$
- $\left\lfloor \dfrac{N}{e} \right\rfloor + 1$
- $\left\lfloor \dfrac{N}{e^2} \right\rfloor$
- $\left\lfloor \dfrac{N}{e^2} \right\rfloor + 1$
So eliminating the last two. we are left with two options. I did a little bit of hit and trial and found out that both the options may be correct according to the value of $N$. However, I am unable to find out a general expression as to what values of $N$ will give a maximum with the second option and which will give it with the first.
And also the fact as to why $N$ is greater than or equal to $5$. Maybe, because with $N =4$, both the first and the second options give the same answer? That's just a guess though.