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!

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