Hello everyone, while I was at a leadership conference my mind wandered and I found myself pondering the optimization of functions of the form

So I basically treated it as a function of one variable, and proceeding as such I arrived at a question.

Now by Leibniz's rule the above equation is equivalent to

Now I next looked for critical points.

If we let

Then the critical points occur at four points, namely :

When

Where is undefined

Less obviously, where . This obviously arises from the fact that

And also where is undefined.

Now here is where I ran into an anomaly, I found multiple cases where , satisfies two conatradicting critical point criteria, namely.

is undefined, but .

So this implies that this is a multi-valued function, so what does one do from here? Does one just sweep this doubly critical point under the carpet and continue? I hope not.

Any guidance would be appreciated.

Alex