What is the purpose of defining the notion of inflection point?

Question

What is the purpose of defining inflection point?

I know that it is defined to be the point where the second derivative is zero and the second derivative sign changes.

It has to have some purpose for pure math.

Answer 1

Inflection point is more than just the second derivative being zero as one could take $f(x)=x^4$ which would have the second derivative be zero at $x=0$ yet it isn't an inflection point as the second derivative doesn't change sign. Have you ever looked at a graph of a tangent through an inflection point? $g(x)=x^3$ at $x=0$ would be an example where it is worth noting that the tangent goes through the curve here.

Wikipedia defines it this way:

In differential calculus, an inflection point, point of inflection, flex, or inflection (inflexion) is a point on a curve at which the curvature or concavity changes sign from plus to minus or from minus to plus. The curve changes from being concave upwards (positive curvature) to concave downwards (negative curvature), or vice versa.

I'd imagine this could be useful for considering optimization problems to know if a curve is concave one way or another.