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# Second Derivatives and Beyond

# Points of Inflection

A **point of inflection** or **inflection point**, abbreviated IP, is an *x*-value at which the concavity of the function changes. In other words, an IP is an *x*-value where the sign of the second derivative changes. It might also be how we'd describe Peter Brady's voice.

The usual way to look for inflection points of *f* is to

- find
*f*"

- find all
*x*-values where*f*" is zero or undefined, and

- check each such
*x*-value to see if the sign of*f*" changes there.

Again, we can use graphs to check our work. An inflection point where the function goes from concave up to concave down looks something like this:

An inflection point where the function goes from concave down to concave up looks something like this:

While any point at which *f *' is zero or undefined is a critical point, a point at which *f *" is zero or undefined is *not* necessarily an inflection point. You can think of the points where *f *" is zero or undefined as possible inflection points, but then you need to check each possible inflection point to see if it's a real inflection point.

**Be Careful:** Because *f *"(*c*) = 0 or is undefined doesn't mean *c* is an inflection point. *f *" must have different signs to either side of *c*.

There are two main ways to figure out what the sign of *f *" is doing to either side of a possible inflection point *c*.