extremal points
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Borax Mine, Boron, CABorax Mine, Boron, CA
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Finding Extremal Points by Graphing
A Lamborghini Diablos gas mileage can be expressed asa function of instantaneous velocity:
3 20.00015 0.032 1.8 1.7m v v v v
At what speed will the the car get the best gas mileage?
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3 20.00015 0.032 1.8 1.7m v v v v
We could solve the problem graphically:
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3 20.00015 0.032 1.8 1.7m v v v v
We could solve the problem graphically:
On a TI-89, we use F5 (math), 4: Maximum, choose lowerand upper bounds, and the calculator finds our answer.
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3 20.00015 0.032 1.8 1.7m v v v v
We could solve the problem graphically:
On the TI-89, we use F5 (math), 4: Maximum, chooselower and upper bounds, and the calculator finds our
answer.
The car will get approximately 32 miles per gallonwhen driven at 38.6 miles per hour.
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3 20.00015 0.032 1.8 1.7m v v v v
Notice that at
the top of thecurve, thehorizontal
tangent has aslope of zero.
Traditionally, this fact has been used both as an aid tographing by hand and as a method to find maximum (andminimum) values of functions.
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Absolute extreme values are either maximum orminimum points on a curve.
They are sometimes called global extrema.
They are also sometimes called absolute extrema.(Extrema is the plural of the Latin extremum.)
Finding Extremal Points by Analysis
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Extreme values can be in the interior or the end points ofa function.
0
1
2
3
4
-2 -1 1 2
2y x
,D Absolute Minimum
No end points,
thus No AbsoluteMaximum
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0
1
2
3
4
-2 -1 1 22y x
0,2D
Absolute Minimum
AbsoluteMaximum
Note: this is a closed interval(endpoints included).
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0
1
2
3
4
-2 -1 1 2
2y x
0,2D
No Minimum
AbsoluteMaximum
This, on the other hand, is a half-openinterval (only one endpoint included).
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0
1
2
3
4
-2 -1 1 22y x
0,2D
No Minimum
NoMaximum
Note: this is an open interval (endpointsexcluded).
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Extreme Value Theorem:
Iff is continuous over a closed interval, then f hasa maximum and minimum value over that interval.
Maximum &
minimumat interior points
Maximum &
minimumat endpoints
Maximum atinterior point,minimum atendpoint
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Local Extreme Values:
A local maximum is the maximum value within someopen interval.
A local minimum is the minimum value within some openinterval.
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Absolute minimum
Local maximum
Local minimum
Absolute maximum
Local minimum
Local extrema are alsocalled relative extrema.
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Local maximum
Local minimum
Notice that extremes in the interior of the function can
occur where is zero or is undefined.f f
Absolute maximum
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Local Extrema:
1. If a function f has a local maximum value or alocal minimum value at an interior pointc of itsdomain, and
2. If exists at c, then 0f c f
?
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Critical Point:
0f
A point in the domain of a functionf
at which
or does not exist is a critical point off.f
Maximum and minimum values in the interior of a functionalways occur at critical points, but critical points are notalways global maximum or minimum values.
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EXAMPLE: FINDING ABSOLUTE EXTREMA
Find the absolute maximum and minimum values ofon the closed interval . 2/3f x x 2,3
2/3f x x Are there values of x that will makethe first derivative equal to zero?
323f x
x
1
323
f x x The first derivative is undefined at x=0,so (0,0) is a critical point.
Because the function is defined over aclosed interval, we also must check theendpoints.
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To determine if this critical point is actually a
maximum or minimum, we test points to either side.
2/3f x x
1 1f 1 1f
Since 0 < 1, (0, 0) must be at least a local minimum, andpossibly a global minimum. Lets test the endpoints:
2,3D
0 0f At: 0x
At: 2x
2
32 2 1.5874f
At: 3x 2
33 3 2.08008f
.0 undefinedisfand
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To determine if this critical point isactually a maximum or minimum, wetry points on either side, without
passing other critical points.
Since 0
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Absolute minimum (0,0)
Absolute maximum (3,2.08)
2/3f x x
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Finding Maxima and Minima Analytically:
1 Find the derivative of the function, and determinewhere the derivative is zero or undefined. Theseare the critical points.
2 Find the value of the function at each critical point.
3 Find functional values for points between thecritical points to determine if the critical points aremaxima or minima.
4 For closed intervals, check the end points aswell.
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Critical points are not always extremal!
-2
-1
0
1
2
-2 -1 1 2
3y x
0f
(not extremal)
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-2
-1
0
1
2
-2 -1 1 2
1/3y x
is undefined.f
(not extremal)
Critical points are not always extremal!