5.2.6 Concavity & Points of Inflection

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
• CONCAVE DOWN is often called concave
• CONCAVE UP is often called convex

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
    • 

• • 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

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

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

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