Extreme Values of Functions - PowerPoint Presentation

Slide1

Extreme Values of Functions

Chapter 5.1Slide2

Absolute (Global) Extreme Values

Up to now we have used the derivative in applications to find rates of change

However, we are not limited to the rate-of-change interpretation of the derivative

In this section you will learn how we can use derivatives to find extreme values of functions (that is maximum or minimum values)

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Definition of Extreme Values on an Interval

DEFINITION:

Let

be defined on an interval

containing

.

is the minimum of on if for all in is the maximum of on if for all in The minimum and maximum of a function on an interval are the extreme values or extrema (plural of extremum), of the function on the interval. The minimum and maximum of a function on an interval are also called the absolute minimum and absolute maximum. They are also sometimes called the global minimum and global maximum.

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Definition of Extreme Values on an Interval

The definition given is slightly different than the one given in your textbook

Most of the time we are only concerned with extreme values

in an interval, rather than on the entire domain (many functions have neither absolute maxima nor absolute minima over the entire domain)

Over an interval, extrema can occur either in the interior or at the endpoints

It is possible for a function to have no extrema on an interval

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Definition of Extreme Values on an Interval

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Definition of Extreme Values on an Interval

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Definition of Extreme Values on an Interval

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Example 1: Exploring Extreme Values

Use the graphs of

and

on

to determine absolute maxima and minima (if any).

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Example 1: Exploring Extreme Values

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The Extreme Value Theorem

THEOREM:

If

is continuous on a closed interval

, then

has both a maximum value and a minimum value on the interval.

 10Slide11

The Extreme Value Theorem

The proof of this theorem requires more advanced calculus, so we will take the theorem as given

Note that continuity is a requirement of the proof; if we know that the function is not continuous on a given interval, then we cannot use the theorem

In plain words, this tells us that a function is

guaranteed

to have both a maximum and a minimum value on a closed interval (if continuous)

These extrema may be either at the endpoints of the interval or the interior of the interval11Slide12

The Extreme Value Theorem

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The Extreme Value Theorem

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The Extreme Value Theorem

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The Extreme Value Theorem

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Local (Relative) Extrema

In addition to absolute (or global) extrema, which are always the greatest/least function value on an interval, we will want to define relative (local) extrema

These occur when “nearby” values are all less (for a relative maximum) or greater (for a relative minimum)

Relative extrema may also be absolute extrema, but not all relative extrema are absolute extrema (but all absolute extrema are also relative extrema)

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Local (Relative) Extrema

DEFINITION:

Let

be an interior point of the domain of the function

. Then

is a

local maximum value at if and only if for all in some open interval containing local minimum value at if and only if for all in some open interval containing A function has a local maximum or local minimum at an endpoint if the appropriate inequality holds for all in some half-open interval containing  17Slide18

Local (Relative) Extrema

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Local (Relative) Extrema

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Relative extrema in the interior of an interval occur at points where the graph of a function changes direction (from increasing to decreasing, or vice versa)

We would like to be able to find both absolute and relative extrema for a function over a closed interval

To narrow down the possibilities for the interior of an interval, we can ask, “is there anything about relative extrema by which we can identity them?”Slide20

Critical Point

DEFINITION:

Let

be a function defined over some interval

. A point

, where

is in the interior of , at which or is not differentiable is called a critical point of .The number in the interval is called a critical number of . 20Slide21

Critical Point

Note that critical points occur where either

is zero

is not differentiable

The next theorem (not in your text and presented without proof) gives us an answer to our previous question

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Relative Extrema Occur Only at Critical Numbers

THEOREM:

If

has a relative minimum or relative maximum at

, then

is a critical number of

(and is a critical point of ). 22Slide23

Relative Extrema Occur Only at Critical Numbers

This theorem joins the definition of a critical number with the definition of relative extrema

Specifically, it says that

relative extrema occur only at critical numbers

Therefore, to find relative extrema, we need only find critical numbers, which occur where either

is zero or where

is not differentiableHowever, we must be clear about what it does not say: it does not say that, if we find a critical number, then we have found a relative extremum 23Slide24

Relative Extrema Occur Only at Critical Numbers

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Finding Absolute Extrema

We now have enough understanding to be able to find the absolute extrema on a closed interval

If we have a closed interval, the Extreme Value Theorem guarantees that there exist both an absolute maximum and absolute minimum in the interval

These absolute extrema may occur at either the endpoints or in the interior

If absolute extrema occur in the interior, then they occur at relative extrema (which, in turn, occur at critical numbers)

To find absolute extrema, we will first find relative extrema in the interior (i.e., find critical numbers), evaluate the function at this

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Finding Absolute Extrema

To find absolute extrema

We will first find relative extrema in the interior (i.e., find critical numbers)

Evaluate the function at all critical numbers

Evaluate the function at its endpoints

Compare these values; the largest of these is the absolute maximum and the smallest is the absolute minimum

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Example 3: Finding Absolute Extrema

Find the absolute maximum and minimum values of

on the interval

.

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Example 3: Finding Absolute Extrema

Find the absolute maximum and minimum values of

on the interval

.

Find the critical numbers by taking the derivative:

Critical numbers occur where

is zero or where is not differentiable. Note that is never zero. However, is not defined at , which means that is not differentiable at , by our definition, is a critical number (and

is a critical point). Now, check the function values at

,

, and

:

The maximum is approximately 2.08 and occurs at

; the minimum is 0 and occurs at

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Example 3: Finding Absolute Extrema

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Example 4: Finding Extreme Values

Find the extreme values of

.

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Example 4: Finding Extreme Values

Find the extreme values of

.

First note that no interval is provided. From the function it is clear that the domain is

. Since this is not a closed interval, we cannot conclude that the function has both an absolute minimum and absolute maximum. We differentiate to find critical numbers:

We have that

if

and this is the only critical number because critical numbers are only

interior

values of an interval. The critical point occurs at

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Example 4: Finding Extreme Values

Find the extreme values of

.

How can we know whether

is a maximum, minimum, or neither? If

, then the denominator of

decreases, so its reciprocal increases. The same is true if . This means that, for all , and by definition this means that we have an absolute minimum at .Is there an absolute maximum? As approaches 2 from the left, the denominator approaches zero, so the function approaches infinity. The same is true as approaches from the right. So this function has no maximum value. 32Slide33

Example 4: Finding Extreme Values

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Example 5: Finding Local Extrema

Find the local extrema of

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Example 5: Finding Local Extrema

Find the local extrema of

This function cannot have absolute extrema since function values continue increasing to the right of 1, and continue decreasing to the left of 1. What about local (relative) extrema? We differentiate

by differentiating the two pieces of the function. But we must determine whether the function is differentiable at

. To do this, find the left- and right-hand derivatives:

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Example 5: Finding Local Extrema

Find the local extrema of

The derivatives are different, so

is not differentiable at

.

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Example 5: Finding Local Extrema

Find the local extrema of

So our derivative is

Therefore we have critical points at

and

. Since the function piece defined for

is a parabola that opens down, then

must be a local maximum.

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Example 5: Finding Local Extrema

We can determine how to classify the critical point

by examining what happens to the function values at

on either side of (but near)

. To the left of

, note that

, , ; in general, these nearby values are greater than 3. To the right of : , , ; in general, these nearby values are greater than 3. So we have, that for values near

so by definition,

occurs at a local minimum.

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Example 5: Finding Local Extrema

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Example 6: Finding Local Extrema

Find the local extrema of

.

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Example 6: Finding Local Extrema

Find the local extrema of

.

The textbook asks you to use a calculator, but differentiating this function is not beyond your ability if you first note the following:

Suppose that

is a differentiable function of

. Then can be written as a piecewise defined functionThen we get

if

and

, if

. We can ignore the absolute value sign!

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Example 6: Finding Local Extrema

Find the local extrema of

.

Now, if

, then

Find

(with respect to ):

Therefore,

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Example 6: Finding Local Extrema

Find the local extrema of

.

Find where

is zero and where

is not differentiable. The function is not differentiable at

, but this doesn’t count because

is not defined (so zero is not in the domain of

). We have

. The function values for these critical numbers are

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Example 6: Finding Local Extrema

Find the local extrema of

.

Finally, check some nearby function values to determine whether these are local maxima, minima, or neither.

At

:

, ; this appears to be a local maximumAt : , ; this, too, appears to be a local maximum. 44Slide45

Example 6: Finding Local Extrema

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Exercise 5.1

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