1. Extremum & Inflection

2. Finding and Confirming the Points of Extremum & Inflection

3. InflectionExtremum Candidate Points for Inflection Points of the domain at which the second derivative is equal 0 or does not exist. Critical Points Points of the domain at which the first derivative is equal 0 or does not exist. I The second Derivative Test for inflection Testing the sign of the second derivative about the point The First Derivative Test for Local Extremum Testing the sign of the first derivative about the point II The Second Derivative Test for Local Extremum Testing the sign of the second derivative at the point III

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. Extremum The Second Derivative Test for Local Extremum Only for critical points ( x 0, f((x 0 ) ) for which the first derivative f’ (x 0 ) = 0 (1) If f’’ (x 0 ) is negative, then the point ( x 0, f(x 0 ) ) is a point of local maximum (2) If f’’ (x 0 ) is positive, then the point ( x 0, f(x 0 ) ) is a point of local minimum III.

6. Inflection Candidate Points We find candidate points, which include any point (x 0,y 0 ) of f at which either the second derivative f’’ (x 0 ) equal 0 or does not exist The Second Derivative Test for Inflection For each candidate point ( x 0, f((x 0 ) ) we examine the sign of the second 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 inflection, at which: (1) The graph of f changes from concave upward to concave downward if the sign of second derivative f’’ (x) changes from positive on the immediate left to negative on the immediate right of the candidate point x 0 (2) The graph of f changes from concave downward to concave upward if the sign of second derivative f’’ (x) changes from negative on the immediate left to positive on the immediate right of the candidate point x 0 I II.

7. Example Let: f(x) = 2x 3 – 9x 2 +12x + 1 Determine all points of extremum and inflection of the function f and use this information and other information to sketch its graph.

8. Solution 1. Finding Critical Points

9. 2. Intervals of Increase and Decrease of f

10. 3. The First Derivative Test

12. 4. Using the Second Derivative Test for Extremum

13. 5. Intervals of concavity and Point(s) of inflection

15. Graphing f

16. Graph of f

17. f’f’

18. Graphing g=f ’

19. Graph of g=f ’

20. f ’’

21. f ’’’

22. The relation between f and f ’

23. The relation between f and f ’’

25. Try shoving these problems Determine the intervals of increase/decrease and concavity and all points of extremum and inflection of the following functions:

26. For each of the functions f of the previous slide a. Determine the intervals on which f is increasing or decreasing. b. Determine the intervals on which f is concave upward or concave downward.