1. Relative Extrema

2. Objective
To find the coordinates of the relative extrema of a function.

3. Relative Extrema
Relative (local) extrema: points at which a function changes from increasing to decreasing, or from decreasing to increasing

4. Relative Extrema Two Types of Relative Extrema Relative maxima
Relative minima

5. Relative Extrema y Relative maximum Increasing Increasing Decreasing
Relative minimum x
Relative extrema must occur at critical points of the function.

6. Critical Points
Critical points are the places on a graph where the derivative equals zero or is undefined.
Interesting things happen at critical points.

7. Critical Points
Critical points are candidates for the location of maxima and minima of the function.

8. Relative Minimum Relative minimum Slope is Slope is positive.
negative.
Slope is
positive.

9. Relative Maximum Slope is Slope is positive. negative. Relative

10. Neither a Max nor a Min Neither a max nor a min Slope is Slope is
positive.
Slope is
negative.
Slope is
positive.
Slope is
negative.
Neither a
max nor a min

11. The First Derivative Test
If f ’(x) is negative to the left of a critical point and positive to the right, then the critical point is a relative minimum.
If f ’(x) is positive to the left of a critical point and negative to the right, then the critical point is a relative maximum.
If f ’(x) has the same sign to the left and right of a critical point, then the critical point is not a relative extremum.

12. Relative Extrema
Find all relative extrema of the function:

13. Relative Extrema