Question

QUICK QUESTION! Do Hyperbolas have a point of inflection? It has an asymptote so I am confused

Answer 1

In differential calculus, an inflection point, point of inflection, flex, or inflection (inflexion) is a point on a curve at which the curve changes from being

Answer 2

yeah but it seems with a hyperbola that it changes at 2 points at one time...

Answer 3

i dont think conics have inflection points. i dont see that they change their concavity.

Answer 4

maybe if they are rotated so to lie off the graph lines maybe, they might have inflection points as means of referencing their orientation ...

Answer 5

Take the general equation of a hyperbola:
\[\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\]
Take the first derivative:
\[\frac{2x}{a^2}+\frac{2y}{b^2}\frac{dy}{dx}=0~~\iff~~\frac{dy}{dx}=-\frac{b^2x}{a^2y}\]
Take the second derivative:
\[\begin{align*}\frac{d^2y}{dx^2}&=-\frac{b^2}{a^2}\left(\frac{y-x\dfrac{dy}{dx}}{y^2}\right)\\\\
&=-\frac{b^2}{a^2}\left(\frac{y+x\dfrac{b^2x}{a^2y}}{y^2}\right)\\\\
&=-\frac{b^2}{a^2}\left(\frac{a^2y^2+b^2x^2}{a^2y^3}\right)
\end{align*}\]
Does the first derivative have any valid critical points?