Extreme Value Theorem - Endpoint Extremes

1. The Extreme Value Theorem

If a function is continuous on a closed interval then is guaranteed to have global maximum and global minimum values on the interval This is known as the extreme value theorem.

Here is an illustration of the extreme value theorem:

• The function is continuous on the interval
• The maximum point occurs inside the interval.
• The minimum occurs at an endpoint.

f   Continuous Open Interval	f   Continuous Open Interval	f   Not Continuous Closed Interval

term to know

Extreme Value Theorem

If is a continuous function on some closed interval then has global maximum and global minimum values on the interval

2. Finding Extreme Values of a Continuous Function on a Closed Interval

As a result of the theorem, here is what we need to do in order to find the global minimum and maximum values of on a closed interval

1. Find all critical numbers of that are in the interval
2. Evaluate at each endpoint and each critical number. The largest value of f is the global maximum and the smallest value of f is the global minimum.

EXAMPLE

Find the global maximum and minimum points of the function on the interval

First, find the critical numbers.

	Start with the original function.
	Take the derivative.
	Since is a polynomial, it is never undefined. Set and solve for x.

Therefore, the critical numbers are and

However, since only the closed interval is considered, the critical value is not used.

Now, evaluate at the endpoints, and and the remaining critical number,

x		Result
-1		Neither a Global Maximum or Global Minimum
0		Global Maximum
3		Global Minimum

In conclusion, the global maximum occurs at the point (0, 5) and the global minimum occurs at the point (3, -22). In order words, the global maximum value is 5 and occurs when and the global minimum value is -22 and occurs when

The graph of the function on is shown below, which confirms the results.

watch

In this video, we’ll find the global minimum and maximum values of on the interval

When finding critical numbers, it’s important to consider only those that are in the interval Here is an example that helps illustrate this.

EXAMPLE

Find all global extreme values of on the interval

To start, we find the critical numbers.

	Start with the original function.
	Use the power rule to find the derivative.
	Set
	Factor out
	Set each factor equal to 0.
	Solve each equation.

Thus, the critical numbers are and

Considering the interval we are interested in, notice that is not contained in this interval. This means that is not considered in any further analysis.

To determine the values of the local minimum and maximum values, we consider along with the endpoints of the interval, and

Here is a table of values for Note the approximations are also provided to make comparisons easier.

x	1	2	4
f  (x  )

The global minimum value is when and the global maximum value is when

summary

In this lesson, you learned that when is continuous on a closed interval, the extreme value theorem guarantees a global minimum value and a global maximum value at some location within the closed interval. Then, you applied this theorem to find extreme values of a continuous function on a closed interval, by first finding all critical numbers of that are in the interval , then evaluating at each endpoint and each critical number. This concept is going to be very useful once we use derivatives to solve optimization problems.