Suppose we have some function $f(x)$ with local extrema at $x_1, x_2, \dots$, and a second function $g(x)$ which is continuous, strictly increasing and non-zero everywhere over the range of the $x_i$. Will $g(f(x))$ have its local extrema at the same $x_i$ and no others?
If so, are there any obvious loosenings of the constraints on $g$ for which this will remain true?
(I'm really thinking of this in the context of signal processing, looking at transformations that preserve the visual structure of an image, but it seems like a general question that must have been trivially proved by someone 250 years ago...)