The following characteristics are true of all functions and their respective first derivatives:
  • Any x-value that is a zero for f ' corresponds to a relative maximum or minimum for f
  • If f is decreasing then f ' will be negative
  • If f is increasing then f ' will be positive
  • Points of inflection on f are extrema on f '
  • Y values of f ' are the slopes of f
  • Slopes of straight lines on f are constant on f '
  • f ' does not exist when there is a sharp turn or a discontinuity for f


Example: Describe the follwing equation. Be sure to discuss when f' will be decreasing or increasing, points of inflection of the graph, and relative minimums and maximums for the graph of f.

1. f ', when is it decreasing or increasing
- Because the slope of the original function is decreasing from the x value negative infinity to -1.5 f ' will be negative on this same interval.
- Because the slope of the original function is increasing from -1.5 to infinity f ' will be positive on this same interval.
2. Points of Inflection
- Because POI's are extrema for f ' we can determine that there are no points of inflection, because there are no extrema for f '.
3. Relative maximums and minimums
- Because f '=2x+3 we can conclude that there will be a max or min at x= -3/2 (because this is the zero of f '). To determine whether this is a max or min just plug values before and after into f ' to find whether the slope is positive or negative. By doing this we find that the slope is negative before -3/2 and positive after. This means that there is a relative minimum at x= -3/2.


AP Example