Find the maximum value

[cold plover]

Given x,y as positive real integers and 2x + xy = 4.
Find the max value of A = (x^2)y.

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[tight otter]

Note that x^2 y = x(xy).
So, you can express xy from the given constraint, then you'll get a function of a single variable.

[cold plover]

wdym?

is there another solution that doesnt involve functions?

[tight otter]

What do you mean? We are looking for the maximum value of the function A(x, y) = x^2 y.

How can you find the maximum value of a function without using functions?

[cold plover]

alr then but how do i get max(a)?

[tight otter]

As I said, we have A = x(xy). We can express xy from the constaint:
2x + xy = 4
xy = 4 - 2x
So, we get A = x(4 - 2x), which is now easy to analyze.

[cold plover]

ummmmm

how can we proceed from there?

im a newbie in this approach

[tight otter]

Well, it's a quadratic function.

Its extremum is at its vertex.

[cold plover]

is there any solution that is similar to this -(a+b)^2 + c <= c?

[tight otter]

Oh, you want to complete the square.