In how many parts an integer `N>=5` should be dissected so that the product of the parts is maximized.

Explanation

`(x_(1) + x_(2) + ….. + x_(n))/(n) ge (x_(1)x_(2) …… x_(n))^(1//n)`
or `x_(1) x_(2) …… x_(n) le ((x_(1) + x_(2) + …… + x_(n))/(n))^(n)`
Therefore, maximum value of `x_(1) x_(2) x_(3)…….x_(n)` is obtained when `x_(1) = x_(2) = x_(3) = …… x_(n),` i.e., the parts are all equal. Now,
`x_(1) + x_(2) + x_(3) + ...... + x_(n) = N`
Now, function to be maximized is `((x_(1) + x_(2) + ..... x_(n))/(n))^(n)` which is a discrete function of n.
In order to arrive at some possible neighborhood, we make it continous first. Thus, changing the variable n to x, we have
`f (x) = ((X).(x))^(x)`
For maxima, f'(x) = 0 i.e.,
`f'(x) = f(x) (In ((N)/(x))-1)`
`:. f'(x) = 0` for x = N/e
Hence, the nearest integer is [N/e] or [N/e] + l, where [x] denotes the greatest integer less than or equal to x.