Second Derivatives and Beyond
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.