Relative Extrema

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
Decreasing
Increasing
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 positive.
Slope is negative.
9
Relative Maximum
Slope is positive.
Slope is negative.
Relative maximum
10
Neither a Max nor a Min
Neither a max nor a min
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
0
0