Identify points of inflection from a graph

An inflection point is a point on a function’s graph where the concavity changes—typically from concave up, with slopes increasing, to concave down, with slopes decreasing, or vice versa—and the point’s coordinates can be identified from the graph. The understanding distinguishes a genuine change in concavity from an ordinary bend or a local maximum or minimum; generalized or higher-dimensional notions of inflection are not included.

Detailed Explanation: Identify points of inflection from a graph

To identify an inflection point from a graph, look for where the graph changes concavity:

Concave up: the slopes are increasing; the graph bends like $\cup$ .
Concave down: the slopes are decreasing; the graph bends like $\cap$ .

A point is an inflection point only if the concavity changes from one side of the point to the other. A local maximum or minimum is not automatically an inflection point.

Example

Suppose the graph of $f$ has:

a local maximum at $(-1,4)$ ,
a local minimum at $(1,0)$ ,
and a noticeable change in the way it bends at $(0,2)$ .

To find the inflection point:

Start to the left of $(0,2)$ .
The graph bends downward, so it is concave down.
Start to the right of $(0,2)$ .
The graph bends upward, so it is concave up.
Since the concavity changes from concave down to concave up at $x=0$ , the graph has an inflection point there.

Therefore, the inflection point is

\boxed{(0,2)}.

The points $(-1,4)$ and $(1,0)$ are local maximum and minimum points, respectively. They show a change in whether the function is increasing or decreasing, but they do not represent the required change in concavity.

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Critical Points - Function and Second Derivative to Inflection Point Critical Point

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