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Mathematical Expression Editor

We examine a fact about continuous functions.

(a)

A function has a global maximum at , if for every in the domain of
the function.

(b)

A function has a global minimum at , if for every in the domain of
the function.

A global extremum is either a global maximum or a global minimum.

Let be the function given by the graph below.

Find the -coordinate of the point where the function has a global maximum.

Observe that for all in the domain of . Notice, that the function has also a local
maximum at .

Find the -coordinate of the point where the function has a global minimum.
Observe that for all in the domain of . Notice, that the function does not have a
local minimum at . Recall, a function cannot not have a local extremum at a
boundary point.

Find the -coordinate(s) of the point(s) where the function has a local minimum.
Observe that for all in the interval . But it is not true that for all in the domain of
. For example, .

Does every function attain a global extremum on its domain? Select the correct
answer.

All functions must attain both global minimum and global maximum on their
domain.All functions attain a global minimum on their domain.All functions
attain a global maximum on their domain.Some functions have no global
extremums on their domain.

Check the following graph.

Notice, the function is not continuous at , and, therefore, is not continuous on
its domain, . Does the function attain a global extremum on its domain? Select the correct answer.

The function attains both global minimum and global maximum on its domain.The function attains a global minimum, but has no global maximum on
its domain.The function attains a global maximum, but has no global
minimum on its domain.The function has no global extremum on its domain.

Check the following graph.

Notice, the function is continuous on its domain . Does the function given by the
graph above attain a global extremum on its domain? Select the correct answer.

The function attains both global minimum and global maximum on its domain.
The function attains a global minimum, but has no global maximum on
its domain.The function attains a global maximum, but has no global
minimum on its domain.The function has no global extremum on its domain.

Check the following graph.

Notice, the function is continuous on a closed interval . Does the function given
by the graph above attain a global extremum on its domain? Select the correct
answer.

The function attains both global minimum and global maximum on its domain.The function attains a global minimum, but has no global maximum on
its domain.The function attains a global maximum, but has no global
minimum on its domain.The function has no global extremum on its domain.

Find the x-coordinate(s) of the point(s) where the function has a global minimum.

Find the x-coordinate(s) of the point(s) where the function has a global maximum.

Sometimes it is important to know whether a function attains a global
extremum on its domain. The last three examples suggest the following
theorem.

Extreme Value Theorem If is a continuous function for all in the closed interval ,
then there are points and in , such that is a global maximum and is a global
minimum on .

Below, we see a geometric interpretation of this theorem.

Would this theorem hold if we were working on an open interval?

yesno

Consider for . Does this function achieve its maximum and minimum?

Would this theorem hold if we were working on a closed interval , but a function is
not continuous on ?

yesno

Consider a function on a closed interval , defined by for and . Does this function
achieve its maximum and minimum?

Assume that a function is continuous on a closed interval . By the Extreme Value
Theorem, attains both global extremums on the interval . How can we locate these
global extrema? We have seen that they can occur at the end points or in the
open interval . If a global extremum occurs at a point in the open interval ,
then has a local extremum at . That means that has a critical point at .
So, the global extrema of a function occur either at the end points, or ,
or at critical points. If we want to locate the global extrema, we have to
evaluate the function at the end points and at critical points, and compare the
values.

Find the extreme values of on the interval .

The derivative is

The critical points are and (both these lie in ), with values and .

The endpoints have values and .

That gives us a list of values .

The global maximum value of is (occurring at ) and a global minimum value of
(occurring at and ).

Let , for . Locate the global extremums of on the closed interval . Does
the function satisfy the conditions of the Extreme Value Theorem on its
domain?

Yes, because is continuous on .Yes, because is continuous on .No, because is not continuous on .No, because is not differentiable on .

Therefore, the Extreme Value Theorem guarantees that the function attains both
global extremums on its domain. The global extremums occur at the end points or at
critical points.

Find the critical points of . First, compute the derivative of .
In order to find the critical points of , we have to solve the equation
It follows that the function has only one critical point . Find .
In order to locate the global extremums of , we have to evaluate at the end points
and at the critical point.

Order the three values, , , and , from smallest to largest. You should replace with its
value, when you write in your answer below.

Based on this comparison, find the location of the global minimum and global
maximum of . Circle the correct answer.

has a global maximum at and
global minimum at . has a global maximum at and global minimum at . has a global maximum at and global minimum at . has a global
maximum at and global minimum at . has a global maximum at and
global minimum at . has a global maximum at and global minimum at .

Find the extreme values of on .

To find critical points:

Notice that -values with have , so . These values are not critical points.

In the interval , the solutions of are . (The solutions not in are ignored.)

The endpoints have values

The global maximum value of is (occurring at ) and a global minimum value of
(occurring at ).