Concavity and Inflection Points

In this section, you will see how locating the intervals in which the derivative of a function increases or decreases can be used to determine where the graph of a function is curving upward or curving downward.

  • If an interval is curving (or bending) upward, it is considered concave upwards. Its second derivative along this interval will be positive.
  • If an interval is curving (or bending) downward, it is considered concave downwards. Its second derivative along this interval will be negative.
  • If an interval is not curving at all, then its second derivative is zero.

Essential Questions

How are derivatives used to determine where a function is concave upward, concave downward, or neither?