# 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

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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

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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