Relative ExtremaRelative Extrema

ObjectiveObjective

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

Relative ExtremaRelative Extrema

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

Relative ExtremaRelative Extrema

Two Types of Relative ExtremaTwo Types of Relative Extrema1.1. Relative maximaRelative maxima

2.2. Relative minima Relative minima

Relative ExtremaRelative Extrema

Relative extrema must occur at critical points of the function.

x

y

Relative maximum

Relative minimum

Decreasing Incr

easing

Incr

easing

Critical PointsCritical Points

Critical points are the places on a Critical points are the places on a graph where the derivative equals graph where the derivative equals zero or is undefined. zero or is undefined.

Interesting things happen at critical Interesting things happen at critical points.points.

Critical PointsCritical Points

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

Relative MinimumRelative Minimum

Slope isnegative.

Slope ispositive.

Relativeminimum

Relative MaximumRelative Maximum

Slope isnegative.

Slope ispositive.

Relativemaximum

Neither a Max nor a MinNeither a Max nor a Min

Slope ispositive.

Slope ispositive.

Neither amax nor a min

Slope isnegative.

Slope isnegative.

Neither amax nor a min

The First Derivative TestThe First Derivative Test

1.1. If If f f ’(’(xx) ) is negative to the left of a critical is negative to the left of a critical point and positive to the right, then the point and positive to the right, then the critical point is a relative minimum.critical point is a relative minimum.

2.2. If If f f ’(’(xx) ) is positive to the left of a critical is positive to the left of a critical point and negative to the right, then the point and negative to the right, then the critical point is a relative maximum.critical point is a relative maximum.

3.3. If If f f ’(’(xx) ) has the same sign to the left and has the same sign to the left and right of a critical point, then the critical right of a critical point, then the critical point is not a relative extremum.point is not a relative extremum.

Relative ExtremaRelative Extrema

Find all relative extrema of the function:Find all relative extrema of the function:3 2( ) 2 3 12 4f x x x x

2'( ) 6 6 12f x x x 20 6( 2)x x

0 6( 1)( 2)x x

1 0x 2 0x 1x 2x

Relative ExtremaRelative Extrema

'( )f x1 2

0 0

2x '( 2) 0f

'( ) 6( 1)( 2)f x x x

0x '(0) 0f

3x '(3) 0f

Relative MaximumAt 1 x

Relative MinimumAt 2 x

( 1, )

(2, )

11

16

3 2( ) 2 3 12 4f x x x x