I wonder if there is a transformation between functions whose extremum are same, finding extremum of general function becomes easy a bit, but is it known already?
For example, $f(x) = x^2$ has its extrema at $\langle x \rangle = 0$. It is so simple already, but we know there are lots of function whose extrema is at same point. $f(x) = a x^2 + c$, $f(x) = ax^3 + c$ and $f(x) = \frac{a}{4} x^4 + \frac{b}{3} x^3 + \frac{c}{2} x^2 + d$ with $b^2 - 4ac < 0$ are minimum examples, I think. There are infinite examples, of course.
However, as a matter of practice, we would like to know whether an extrema is stable or not. This may add restriction. In the previous expample, $f(x) = ax^3 + c$ is not stable at $\langle x \rangle = 0$.
This amounts to a problem that finding functions whose derivatives have the same set of their roots and values of their derivative at the roots are positive.
I think maybe this is trivial in some areas like Algebraic Geometry, but I don't major in mathematics.
If you know something, please tell me.