1. Simple Tests for Extreme Points

2. Objectives Students will be able to Find absolute maximum and minimum points of a function.

3. First-Derivative Test for Maximum/Minimum If f’ (x) ≥ 0 for x ≤ c and f’ (x) ≤ 0 for x ≥ c, then x = c is a maximum point for the function f. AND If f’ (x) ≤ 0 for x ≤ c and f’ (x) ≥ 0 for x ≥ c, then x = c is a minimum point for the function f.

4. Maximum/Minimum for Concave Down/Up Functions Suppose the function f is concave down/up in on interval I. If x = c is a critical point for the function f in the interior of the interval I, then c is a maximum/minimum point for f in I.

5. Example 1 Find the locations and values of all absolute extrema for the function with the graph

6. Example 2 Find the locations and values of all absolute extrema for the function with the graph

7. Example 3 Find the locations and values of all absolute extrema for the function with the graph

8. Example 4 Find the critical points for the function below and determine if they are absolute maximum or minimum points or neither.

9. Example 5 Find the critical points for the function below and determine if they are absolute maximum or minimum points or neither.

10. Example 6 Find the critical points for the function below and determine if they are absolute maximum or minimum points or neither.

11. Example 7 The height of a flowering plant after t months is given by At what time is the plant at it highest?