We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.
© 2016 Shmoop University, Inc. All rights reserved.
Second Derivatives and Beyond

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 "
  • find all x-values where " is zero or undefined, and
  • check each such x-value to see if the sign of " 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 ' is zero or undefined is a critical point, a point at which " is zero or undefined is not necessarily an inflection point. You can think of the points where " 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: Just because "(c) = 0 or is undefined doesn't mean c is an inflection point. " must have different signs on either side of c.

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

People who Shmooped this also Shmooped...