3.2: Extrema and the Mean Value Theorem
Learning Objectives
A student will be able to:
- Solve problems that involve extrema.
- Study Rolle’s Theorem.
- Use the Mean Value Theorem to solve problems.
Introduction
In this lesson we will discuss a second application of derivatives, as a means to study extreme (maximum and minimum) values of functions. We will learn how the maximum and minimum values of functions relate to derivatives.
Let’s start our discussion with some formal working definitions of the maximum and minimum values of a function.
- Definition
- A function has a maximum at if for all in the domain of Similarly, has a minimum at if for all in the domain of The values of the function for these values are called extreme values or extrema.
- Here is an example of a function that has a maximum at and a minimum at :
- Observe the graph at . While we do not have a minimum at , we note that for all near We say that the function has a local minimum at Similarly, we say that the function has a local maximum at since for some contained in open intervals of
- Let’s recall the Min-Max Theorem that we discussed in lesson on Continuity.
Min-Max Theorem: If a function is continuous in a closed interval then has both a maximum value and a minimum value in In order to understand the proof for the Min-Max Theorem conceptually, attempt to draw a function on a closed interval (including the endpoints) so that no point is at the highest part of the graph. No matter how the function is sketched, there will be at least one point that is highest.
We can now relate extreme values to derivatives in the following Theorem by the French mathematician Fermat.
Theorem: If is an extreme value of for some open interval of and if exists, then
Proof: The theorem states that if we have a local max or local min, and if exists, then we must have
Suppose that has a local max at Then we have for some open interval with
So
Consider .
Since , we have
Since exists, we have , and so
If we take the left-hand limit, we get
Hence and it must be that
If is a local minimum, the same argument follows.
- Definition
- We will call a critical value in if or does not exist, or if is an endpoint of the interval.
- We can now state the Extreme Value Theorem.
Extreme Value Theorem: If a function is continuous in a closed interval , with the maximum of at and the minimum of at then and are critical values of
Proof: The proof follows from Fermat’s theorem and is left as an exercise for the student.
Example 1:
Let’s observe that the converse of the last theorem is not necessarily true: If we consider and its graph, then we see that while at is not an extreme point of the function.
Rolle’s Theorem: If is continuous and differentiable on a closed interval and if then has at least one value in the open interval such that .
The proof of Rolle's Theorem can be found at http://en.wikipedia.org/wiki/Rolle's_theorem.
Mean Value Theorem: If is a continuous function on a closed interval and if contains the open interval in its domain, then there exists a number in the interval such that
Proof: Consider the graph of and secant line as indicated in the figure.
By the Point-Slope form of line we have
and
For each in the interval let be the vertical distance from line to the graph of Then we have
for every in
Note that Since is continuous in and exists in then Rolle’s Theorem applies. Hence there exists in with
So for every in
In particular,
and
The proof is complete.
Example 2:
Verify that the Mean Value Theorem applies for the function on the interval
Solution:
We need to find in the interval such that
Note that and Hence, we must solve the following equation:
By substitution, we have
Since we need to have in the interval the positive root is the solution, .
Lesson Summary
- We learned to solve problems that involve extrema.
- We learned about Rolle’s Theorem.
- We used the Mean Value Theorem to solve problems.
Multimedia Links
For a video presentation of Rolle's Theorem (8.0), see Math Video Tutorials by James Sousa, Rolle's Theorem (7:54).
For more information about the Mean Value Theorem (8.0), see Math Video Tutorials by James Sousa, Mean Value Theorem (9:52).
For a well-done, but unorthodox, student presentation of the Extreme Value Theorem and Related Rates (3.0)(12.0), see Extreme Value Theorem (10:00).
Review Questions
In problems #1–3, identify the absolute and local minimum and maximum values of the function (if they exist); find the extrema. (Units on the axes indicate ).
- Continuous on
- Continuous on
- Continuous on
In problems #4–6, find the extrema and sketch the graph.
- Verify Rolle’s Theorem by finding values of for which and
- Verify Rolle’s Theorem for
- Verify that the Mean Value Theorem works for
- Prove that the equation has a positive root at and that the equation has a positive root less than
Review Answers
- Absolute max at absolute minimum at relative maximum at Note: there is no relative minimum at because there is no open interval around since the function is defined only on the extreme values of are
- Absolute maximum at absolute minimum at relative minimum at Note: there is no relative minimum at because there is no open interval around since the function is defined only on the extreme values of are
- Absolute minimum at ; there is no maximum since the function is not continuous on a closed interval.
- Absolute maximum at absolute minimum at
- Absolute maximum at absolute minimum at
- Absolute minimum at
- at (by Rolle’s Theorem, there is a critical value in each of the intervals and and we found those to be )
- at at by Rolle’s Theorem, there is a critical value in the interval and we found it to be
- Need to find such that
- Let Observe that By Rolle’s Theorem, there exist such that