relative extrema. objective to find the coordinates of the relative extrema of a function. to find...
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Relative ExtremaRelative Extrema
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ObjectiveObjective
To find the coordinates of the relative To find the coordinates of the relative extrema of a function.extrema of a function.
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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
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Relative ExtremaRelative Extrema
Two Types of Relative ExtremaTwo Types of Relative Extrema1.1. Relative maximaRelative maxima
2.2. Relative minima Relative minima
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Relative ExtremaRelative Extrema
Relative extrema must occur at critical points of the function.
x
y
Relative maximum
Relative minimum
Decreasing Incr
easing
Incr
easing
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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.
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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.
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Relative MinimumRelative Minimum
Slope isnegative.
Slope ispositive.
Relativeminimum
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Relative MaximumRelative Maximum
Slope isnegative.
Slope ispositive.
Relativemaximum
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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
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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.
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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
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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