If the numbers may be identical, 50 and 50 => product is 2500

If the numbers must be different, 49 and 51 => product is 2499

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The sum of two positive real numbers is 100. Find their maximum possible product?

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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.

...

d(100x-x^2)/dx=0

x=50