Somehow the piece of information that the initial N = 5 when t = 0 was not very useful unless it was for other parts of the question

As the other comment said, it is wrong to write it as k(N-N^2) as they do not necessarily have the same proportionality constant. Therefore, I wrote the proportionality constants as alpha and beta, so our task will be to find values of alpha and beta by basically forming 2 different equations

Essentially what you do is firstly, when they say it eventually reaches 100, that simply means when dN/dt = 0, u can take N as 100 which will allow you to express alpha in terms of beta.

The next thing you do is with the number 10. You can intepret it as when they say the max dN/dt it means that it is the maximum point (turning point) on the dN/dt graph therefore the d2N/dt2 will be 0. then you will also have to basically express N at this point in terms of beta and then substitute it back into the equation to find beta.