I was studying if the curvature of $f$: $$ f(x) = \frac{ax}{b+x} $$

can have the critical points located at the same vertical than this other $g$ curve:

$$ g(x) = \frac{ax}{b+x} + cx = f(x) + cx $$

Then I've realised that $g(x)$ actually is a linear transformation of $f(x)$, and I thought that maybe this linear transformation (a shear) would not alter the vertical position of the critical points of the curvature just by intuition. Sadly, I was wrong... But a more broad question came to my mind, and I think it can be worthy to share it with you:

What is the curve that keeps at least one coordinate fixed of the critical points of its curvature invariant under a shear linear transformation?

Extra1: and for other linear transformations?

Extra2: what if we want to keep both coordinates of the critical point fixed?

I know, very broad question, maybe. But I don't even know where I should start!

Many thanks in advance!!


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