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Second Derivatives and Beyond

Second Derivatives and Beyond

At a Glance - 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.

Example 1

Find all points of inflection for the function f (x) = x3.


Example 2

Find all inflection points for the function f (x) = x4.


Example 3

Suppose we started with a function f, took a couple of derivatives, and found that

"(x) = (x + 2)(x + 1).

What are the inflection points of ?


Exercise 1

For the function, find all points of inflection or determine that no such points exist.

f (x) = sin x


Exercise 2

For the function, find all points of inflection or determine that no such points exist.

f (x) = ex


Exercise 3

For the function, find all points of inflection or determine that no such points exist.


Exercise 4

For the function, find all points of inflection or determine that no such points exist.


Exercise 5

For the function, find all points of inflection or determine that no such points exist.

f (x) = 5x + 2


Exercise 6

For the function, find all points of inflection or determine that no such points exist.

f (x) = x1/3


Exercise 7

For the function, find all points of inflection or determine that no such points exist.

f (x) = xex


Exercise 8

For the function, find all points of inflection or determine that no such points exist.


Exercise 9

For the function, find all points of inflection or determine that no such points exist.

f (x) = ln x for x > 0


Exercise 10

For the function, find all points of inflection or determine that no such points exist.

The logistic equation


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