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Mathematical Expression Editor

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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, say, . 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 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 graph below.

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 following theorem, which comes as no surprise after the
previous three examples, gives a simple answer to that question.

Extreme
Value Theorem If is continuous on 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 or 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 or 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.

Let , for .

(a)

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 .

Now we know that the Extreme Value Theorem guarantees that the function
attains both global extremums on its domain!

(b)

Locate the global extremums of on the closed interval .

The global extremums occur at the end points or at critical points.

Let’s 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, the function has the global minimum at , and the
global maximum at .

For your convenience, the graph of on the interval is given below.