1. First derivative: is positive Curve is rising. is negative Curve is falling. is zero Possible local maximum or minimum. Second derivative: is positive Curve is concave up. is negative Curve is concave down. is zero Possible inflection point (where concavity changes).

2. Example: Graph There are roots at and. Set First derivative test: negative positive Possible extreme at. We can use a chart to organize our thoughts.

3. Example: Graph There are roots at and. Set First derivative test: maximum at minimum at Possible extreme at.

4. Example: Graph First derivative test: There is a local maximum at (0,4) because for all x in and for all x in (0,2). There is a local minimum at (2,0) because for all x in (0,2) and for all x in.

5. Because the second derivative at x = 0 is negative, the graph is concave down and therefore (0,4) is a local maximum. Example: Graph There are roots at and. Possible extreme at. Or you could use the second derivative test: Because the second derivative at x = 2 is positive, the graph is concave up and therefore (2,0) is a local minimum.

6. inflection point at There is an inflection point at x = 1 because the second derivative changes from negative to positive. Example: Graph We then look for inflection points by setting the second derivative equal to zero. Possible inflection point at. negative positive

7. Make a summary table: rising, concave down local max falling, inflection point local min rising, concave up 

8. Graph