All the answers I'm seeing are very empirical and seem to be have been arrived at "by experience." Although they are right, the method isn't.
What the problem asks is to maximize x*(100-x). 100x - x2.
Maxima minima (if you can use that) tell us that the maxima is at x=50. Thus, the highest value is 2500.
Otherwise, if you are aware of parabola, you would know that the above expression is a functional representation of a (open-downwards) parabola and you can easily find that the apex exists at x=50. Thus, maximum value is, again, at 2500.