1. Inflection Points

2. Objectives Students will be able to Determine the intervals where a function is concave up and the intervals where a function is concave down. Find all infection points of a function.

3. Definition The point c is called an inflection point for the function f if there exists an interval (a, b) about c such that a.f’’ (x) > 0 in (a, c) and f’’ (x) < 0 in (c, b) or b.f’’ (x) 0 in (c, b)

4. Test for Inflection Points Let f be a function with a continuous second derivative in an interval I, and let c be an interior point of I. a.If c is an inflection point for f, then f’’ (c) = 0. b.If f’’ (c) = 0 and f’’ changes sign at c, then c is an inflection point for f.

5. Example 1 Find the open intervals where the function shown in the graph is concave upward or concave downward. Also indicate any of inflection points.

6. Example 2 Find the open intervals where the function shown in the graph is concave upward or concave downward. Also indicate any of inflection points.

7. Example 3 Find the open intervals where the function is concave upward or concave downward. Find any of inflection points.

8. Example 4 Find the open intervals where the function is concave upward or concave downward. Find any of inflection points.

9. Example 5 Find the open intervals where the function is concave upward or concave downward. Find any of inflection points.

10. Example 6 The graph to the right is the graph of f’ (x), the derivative of f(x). Find the open intervals where the function is concave upward or concave downward. Find any points of inflection.