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 of inflection is a point at which the graph of changes concavity

• Instead 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 at the point

  • 

        AND

• The graph of changes concavity at the point

  •  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 that has a point of inflection at some, say, then

What about the first derivative, like with turning points?

• A point of inflection, unlike a turning point, does not necessarily need to have a first derivative value of 0 (i.e.  is not necessarily true)

  • 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 the concavity changes at , then  is a point of inflection
   

• STEP 3

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