Unformatted text preview: inimum and Maximum Values Many of our applications in this chapter will revolve around minimum and maximum
values of a function. While we can all visualize the minimum and maximum values
of a function we want to be a little more specific in our work here. In particular we
want to differentiate between two types of minimum or maximum values. The
following definition gives the types of minimums and/or maximums values that we’ll
be looking at.
Definition
1. We say that f(x) has an absolute (or global) maximum at
if for every x in the domain we are working on. 2. We say that f(x) has a relative (or local) maximum at if for every x in some open interval around
3. We say that f(x) has an absolute (or global) minimum at
if . for every x in the domain we are working on. 4. We say that f(x) has a relative (or local) minimum at
for every x in some open interval around Note that when we say an “open interval around if
. ” we mean that we can find some interval
, not including the endpoints, such
that
. Or, in other words, c will be contained somewhere
inside the interval and will not be either of the endpoints.
Also, we will collectively call the minimum and maximum points of a function
the extrema of the function. So, relative extrema will refer to the relative minimums
and maximums while absolute extrema refer to the absolute minimums and
maximums.
Now, let’s talk a little bit about the subtle difference between the absolute and relative
in the definition above.
We will have an absolute maximum (or minimum) at
provided f(c) is
the largest (or smallest) value that the function will ever take on the domain that we
are working on. Also, when we say the “domain we are working on” this simply
means the range of x’s that we have chosen to work with for a given problem. There
may be other values of x that we can actually plug into the function but have excluded
them for some reason. A relative maximum or minimum is slightly different. All that’s required for a point
to be a relative maximum or minimum is for that point to be a maximum or minimum
in some interval of x’s around
. There may be larger or smaller values
of the function at some other place, but relative to
, or local to
, f(c) is larger or smaller than all the other function values that are near it.
Note as well that in order for a point to be a relative extrema we must be able to look
at function values on both sides of
to see if it really is a maximum or
minimum at that point. This means that relative extrema do not occur at the end
points of a domain. They can only occur interior to the domain.
There is actually some debate on the preceding point. Some folks do feel that relative
extrema can occur on the end points of a domain. However, in this class we will be
using the definition that says that they can’t occur at the end points of a domain.
It’s usually easier to get a feel for the definitions by taking a quick look at a graph. For the function shown in this graph we have relative maximums at
and
. Both of these point are relative maximums since they are
interior to the domain shown and are the largest point on the graph in some interval
around the point. We also have a relative minimum at
since this point
is interior to the domain and is the lowest point on the graph in an interval around it.
The far right end point,
, will not be a relative minimum since it is an
end point.
The function will have an absolute maximum at
and an absolute
minimum at
. These two points are the largest and smallest that the
function will ever be. We can also notice that the absolute extrema for a function will occur at either the endpoints of the domain or at relative extrema. We will use this
idea in later sections so it’s more important than it might seem at the present time.
Let’s take a quick look at some examples to make sure that we have the definitions of
absolute extrema and relative extrema straight.
Example 1 Identify the absolute extrema and relative extrema for the following function. Solution
Since this function is easy enough to graph let’s do that. However, we only want the graph on the
interval [1,2]. Here is the graph, Note that we used dots at the end of the graph to remind us that the graph ends at these points.
We can now identify the extrema from the graph. It looks like we’ve got a relative and absolute
minimum of zero at
and an absolute maximum of four at
.
Note that
is not a relative maximum since it is at the end point of the
interval.
This function doesn’t have any relative maximums. As we saw in the previous example functions do not have to have relative extrema. It
is completely possible for a function to not have a relative maximum and/or a relative
minimum.
Example 2 Identify the absolute extrema and relative extrema for the following function. Solution Here is the graph for this function. In this case we still have a relative and absolute minimum of zero at
. We also
still have an absolute maximum of four. However, unlike the first example this will occur at two
points,
and
.
Again, the function doesn’t have any relative maximums. As this example has shown there can only be a single absolute maximum or absolute
minimum value, but they can occur at more than one place in the domain.
Example 3 Identify the absolute extrema and relative extrema for the following function.
Solution
In this case we’ve given no domain and so the assumption is that we will take the largest possible
domain. For this function that means all the real numbers. Here is the graph. In this case the graph doesn’t stop increasing at either end and so there are no maximums of any
kind for this function. No matter which point we pick on the graph there will be points both
larger and smaller than it on either side so we can’t have any maximums (or any kind, relative or
absolute) in a graph. We still have a relative and absolute minimum value of zero at . So, some graphs can have minimums but not maximums. Likewise, a graph could
have maximums but not minimums.
Example 4 Identify the absolute extrema and relative extrema for the following function. Solution
Here is the graph for this function. This function has an absolute maximum of eight at
and an absolute minimum of
negative eight at
. This function has no relative extrema. So, a function doesn’t have to have relative extrema as this example has shown.
Example 5 Identify the absolute extrema and relative extrema for the following function.
Solution
Again, we aren’t restricting the domain this time so here’s the graph. In this case the function has no relative extrema and no absolute extrema. As we’ve seen in the previous example functions don’t have to have any kind of
extrema, relative or absolute.
Example 6 Identify the absolute extrema and relative extrema for the following function.
Solution
We’ve not restricted the domain for this function. Here is the graph. Cosine has extrema (relative and absolute) that occur at many points. Cosine has both relative
and absolute maximums of 1 at Cosine also has both relative and absolute minimums of 1 at As this example has shown a graph can in fact have extrema occurring at a large
number (infinite in this case) of points.
We’ve now worked quite a few examples and we can use these examples to see a nice
fact about absolute extrema. First let’s notice that all the functions above were continuousfunctions. Next notice that every time we restricted the domain to a
closed interval (i.e. the interval contains its end points) we got absolute maximums
and absolute minimums. Finally, in only one of the three examples in which we did
not restrict the domain did we get both an absolute maximum and an absolute
minimum.
These observations lead us the following theorem.
Extreme Value Theorem
Suppose that
numbers
for the function and is continuous on the interval [a,b] then there are two
so that is an absolute maximum is an absolute minimum for the function. So, if we have a continuous function on an interval [ a,b] then we are guaranteed to
have both an absolute maximum and an absolute minimum for the function
somewhere in the interval. The theorem doesn’t tell us where they will occur or if
they will occur more than once, but at least it tells us that they do exist somewhere.
Sometimes, all that we need to know is that they do exist.
This theorem doesn’t say anything about absolute extrema if we aren’t working on an
interval. We saw examples of functions above that had both absolute extrema, one
absolute extrema, and no absolute extrema when we didn’t restrict ourselves down to
an interval.
The requirement that a function be continuous is also required in order for us to use
the theorem. Consider the case of Here’s the graph. This function is not continuous at
as we move in towards zero the
function is approaching infinity. So, the function does not have an absolute
maximum. Note that it does have an absolute minimum however. In fact the absolute
minimum occurs twice at both
and
.
If we changed the interval a little to say, the function would now have both absolute extrema. We may only run into problems
if the interval contains the point of discontinuity. If it doesn’t then the theorem will
hold.
We should also point out that just because a function is not continuous at a point that
doesn’t mean that it won’t have both absolute extrema in an interval that contains that
point. Below is the graph of a function that is not continuous at a point in the given
interval and yet has both absolute extrema. This graph is not continuous at
, yet it does have both an absolute
maximum (
) and an absolute minimum (
). Also note
that, in this case one of the absolute extrema occurred at the point of discontinuity, but
it doesn’t need to. The absolute minimum could just have easily been at the other end
point or at some other point interior to the region. The point here is that this graph is
not continuous and yet does have both absolute extrema
The point of all this is that we need to be careful to only use the Extreme Value
Theorem when the conditions of the theorem are met and not misinterpret the results
if the conditions aren’t met.
In order to use the Extreme Value Theorem we must have an interval and the function
must be continuous on that interval. If we don’t have an interval and/or the function
isn’t continuous on the interval then the function may or may not have absolute
extrema.
We need to discuss one final topic in this section before moving on to the first major
application of the derivative that we’re going to be looking at in this chapter.
Fermat’s Theorem
If
then has a relative extrema at
is a critical point of such that exists . In fact, it will be a critical point . Note that we can say that
that and exists. because we are also assuming This theorem tells us that there is a nice relationship between relative extrema and
critical points. In fact it will allow us to get a list of all possible relative extrema.
Since a relative extrema must be a critical point the list of all critical points will give
us a list of all possible relative extrema.
Consider the case of
relative minimum at
Fermat’s theorem
function is, Sure enough . We saw that this function had a
in several earlier examples. So according to
should be a critical point. The derivative of the is a critical point. Be careful not to misuse this theorem. It doesn’t say that a critical point will be a
relative extrema. To see this, consider the following case. Clearly
is a critical point. However we saw in an earlier example this
function has no relative extrema of any kind. So, critical points do not have to be
relative extrema.
Also note that this theorem says nothing about absolute extrema. An absolute extrema
may or may not be a critical point.
To see the proof of this theorem see the Proofs From Derivative
Applications section of the Extras chapter. ...
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This note was uploaded on 11/06/2011 for the course MATH 151 taught by Professor Sc during the Fall '08 term at Rutgers.
 Fall '08
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