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

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

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