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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.

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    $\begingroup$ To analyze them. $\endgroup$
    – MITjanitor
    Mar 7, 2013 at 22:54
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    $\begingroup$ It can be of importance in various ways. For example, if our displacement (position) at time $t$ is given by $s(t)$, the inflection points of $s(t)$ are the places where our acceleration changes from positive to negative, or vice-versa. $\endgroup$ Mar 7, 2013 at 23:05
  • $\begingroup$ @AndréNicolas - I mean the purpose in pure math not physics $\endgroup$
    – Victor
    Mar 7, 2013 at 23:06
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    $\begingroup$ For instance, when there were no beautiful online graphing calculators, it helped drawing graphs properly. Note also that it generalizes to the notion of saddle point in higher dimension: en.wikipedia.org/wiki/Saddle_point This is an important notion in dynamical systems. $\endgroup$
    – Julien
    Mar 7, 2013 at 23:14
  • $\begingroup$ Mathematical physics is hard to separate from mathematics. Here is another example. When we are approximating a function by a secant line (a line joining two points on the curve) it is often useful to know whether we have an overestimate (convex curve, usually called concave up in calculus courses) or an underestimate. Similar considerations apply for approximation by a tangent line. $\endgroup$ Mar 7, 2013 at 23:15

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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.

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