Find the maximum value to this function

Job Description:

The constant product for a stablepool at Solidex Automated Market maker is:

(xy)*(x^2 + y^2) = k

where x is the reserves of token0 and y is the reserves of token1. This constraint must always be upheld when the user performs an exchange. We are interested in finding the change in x that produces the maximum value dy. In more simpler terms; We want to find the amount to sell, so that we will get the maximum amount out. x is reserves and token in, and y is reserves and token out.

We rewrite to include the change in reserves, we sell amount dx and obtain amount dy.

The constraint must hold:

(x + dx)*(y-dy)*((x + dx)^2 + (y - dy)^2) = k

I believe this can also be written as such:

y * (x+dx)^3 + x*(y - dy)^3 = k

It is negative dy and positive dx because the constraint must be upheld for the pool's inventory. So after an exchange, dy has left the pool, and dx has increased the pool.

Here is the task I expect delivery on:

Find the function that provides the maximum dy with respect to dx, x and y as parameters. I require a formula answer, not an algorithmic approach.

In more simpler terms; Which dx maximizes dy? Use dx, x and y as inputs that gives dy as output


Skills: Mathematics, Engineering Mathematics

About the Client:
( 5 reviews ) Nydalen, Norway

Project ID: #35910487

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