Concavity & Points of Inflection (DP IB Analysis & Approaches (AA)): Revision Note

Concavity of a Function

What is concavity?

• Concavity is the way in which a curve (or surface) bends

• Mathematically,

  • a curve is CONCAVE DOWN if for all values of in an interval

  • a curve is CONCAVE UP if for all values of in an interval

Points of Inflection

What is a point of inflection?

• A point at which the curve of the graph of changes concavity is a point of inflection

• The alternative spelling, inflexion, may sometimes be used

What are the conditions for a point of inflection?

• A point of inflection requires BOTH of the following two conditions to hold

• the second derivative is zero

  • 

AND

• the graph of changes concavity

  •  changes sign through a point of inflection

• It is important to understand that the first condition is not sufficient on its own to locate a point of inflection

  • points where could be local minimum or maximum points

    • the first derivative test would be needed

  • However, if it is already known has a point of inflection at, say, then

What about the first derivative, like with turning points?

• A point of inflection, unlike a turning point, does not necessarily have to have a first derivative value of 0 ( )

  • If it does, it is also a stationary point and is often called a horizontal point of inflection

    • the tangent to the curve at this point would be horizontal

  • The normal distribution is an example of a commonly used function that has a graph with two non-stationary points of inflection

How do I find the coordinates of a point of inflection?

• For the function

STEP 1

Differentiate twice to find and solve to find the -coordinates of possible points of inflection

STEP 2    

Use the second derivative to test the concavity of either side of

• If then is concave down

• If then is concave up

If concavity changes,  is a point of inflection

STEP 3

If required, the-coordinate of a point of inflection can be found by substituting the-coordinate into