1. Extremum

2. Finding and Confirming the Points of Extremum

3. Extremum Critical Points Points at which the first derivative is equal 0 or does not exist. I The First Derivative Test for Local Extremum Testing the sign of the first derivative about the point II

4. Extremum Critical Points We find critical points, which include any point ( x 0, f((x 0 ) ) of f at which either the derivative f’ (x 0 ) equal 0 or does not exist. The First Derivative Test for Local Extremum For each critical point ( x 0, f((x 0 ) ) we examine the sign of the first derivative f’ (x) on the immediate left and the immediate right of this point x 0. If there is a change of sign at x 0, then the point ( x 0, f((x 0 ) ) is a point of local extremum, with the extremum being: (1) A local maximum if the sign of f’ (x) is positive on the immediate left and negative on the immediate right of the critical point x 0 (2) A local minimum if the sign of f’ (x) is negative on the immediate left and positive on the immediate right of the critical point I II.

5. Example Let: f(x) = 2x 3 – 9x 2 +12x + 1 Determine all points of extremum of the function f

6. Solution

9. Graph of f

10. Homework (1)

11. Homework (2) For each of the functions f of the previous homework (1) Determine the intervals on which f is increasing or decreasing