# If #y = 1 / (1+x^2)#, what are the points of inflection, concavity and critical points?

First, let's note what each of these mean for a 2D graph on the Cartesian xy-plane.

DEFINITIONS

Concavity is essentially a descriptor noting that on either side of a critical point, the graph's behavior is the same, i.e. the graph increases on both sides, or decreases on both sides.

An inflection point is almost like a point of concavity, but on either side of the critical point, the behavior is opposite. Thus, we call inflection points the point where the concavity changes when moving across the specified critical point.

THE GRAPH ITSELF

Before we start, yes, it kind of spoils where the critical point is, but here's the graph:

graph{1/(1+x^2) [-10, 10, -5, 5]}

CRITICAL POINT

Okay, so there is only one! Great. What is its concavity? Up or down?

CONCAVITY

Using the Quotient Rule and Chain Rule, we get:

INFLECTION POINTS

I'm going to say right upfront---this graph has no inflection points. It only has one extremum---a local maximum, which is concave down. It can't change concavity without becoming a new graph.

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