When reciprocating, local max become local min and vice versa.
Do increasing horizontal points of inflexion remain increasing? Or do they become decreasing.
Edit: Still need confirmation on if this holds in general please. Using GeoGebra I have found that the HPOI does reverse for y=x^3+1
Let's say f is twice continuously differentiable and let y = g(x) := 1/f(x). Then assuming f(a) =/= 0, we have g'(a) = -f'(a)/(f(a)^2). So sign of g'(a) = – (sign of f'(a)).
So anywhere where f would be increasing (f'(x) positive in a neighbourhood of a), g would be decreasing. So increasing places become decreasing.
Note g"(a) = [-f"(a)*(f(a)^2) + 2*f(a)*(f'(a)^2)]/[f(a)^4].
At a point of inflection a for f, we have f"(a) = 0. But in general this won't force g"(a) to be g"(a) = 0 (it would if it were a
horizontal point of inflection).
So the point a may not even be an inflection point for g.
Edit realised you said "increasing horizontal points of inflection". I'd missed your horizontal part because when you said increasing, I automatically thought you meant the slope being either positive or negative at the point.
In the event a is a horizontal point of inflection, then, as said above, g"(a) = 0. Additionally assume f is increasing in some neighbourhood of a, so f'(a) is positive either side of a (positive in some open neighbourhood of a). But I'm not sure yet whether this forces a to be a point of inflection for g. Need to go now but you can investigate this further. I think it should be able to find an example where it's no longer a point of inflection. Try finding one so that g"(x) fails to change sign about a (or equivalently the numerator of g"(x) does, since the denominator is positive). If g"(x) fails to change sign about a, then a won't be an inflection point for g.
In Summary
It's relatively easy to show that g = 1/f will be decreasing where f is increasing, but I'm not sure whether horizontal points of inflection are necessarily preserved under the reciprocal transform.
Edit 2: actually, since a is a POI, f'(x) would maintain sign about a, so so would g'(x), making it a POI for g too.
