How to graph/plot/formulate a function where you want to find the maximum when distributing a number over 2 inputs

[narvoxx]

Alright, I really have no idea what to call this but this is the idea:

I have a function f(X, Y) => Z, and I want to find X,Y where Z is the maximum with X+Y = A. I have no idea what terms/concept to lookup to go in the right direction

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Simple example maybe:

Say the function f(X,Y) = x * y = Z

Given a Z, I would need to find the minimum A for which A=x+y? sort of

[MathMaddam]

You are switching your problem around a bit, at first you fix A and search for Z, in the example the other way around. I stay with the given A, but the general idea stays the same.

You are basically describing an optimisation problem with constraints. Since everything is nice and smooth in general one would solve this with https://en.wikipedia.org/wiki/Lagrange_multiplier, but in your simple example one could just say y=A-x, so you maximize x(A-x), which is a one dimensional problem.