We apply the Mean Value Theorem.

1 Rolle’s Theorem

We begin with a special case of the Mean Value Theorem known as Rolle’s Theorem. This theorem uses the Extreme Value Theorem to guarantee the existence of a critical number for a differentiable function on and interval in which the function is equal at the endpoints.

If is a constant function on , then for all in the open interval and so the theorem is true in this case. If is not constant on the closed interval , the rationale for the verity of the theorem is more complicated. We argue as follows. Since is assumed to be continuous on the closed interval , we can apply the Extreme Value Theorem (section 3.2) to conclude that has absolute an absolute max and absolute min on . Furthermore, the hypotheses of the theorem require that , so the endpoints of a non-constant function cannot be both the max and the min. We have therefore deduced that , must have at least one of its max or min in the open interval . By Fermat’s Theorem (section 3.2), this absolute extreme occurs at a critical number, . Finally, since is differentiable on (by hypothesis), we can conclude that . Thus, the theorem is true in this case as well, and Rolle’s Theorem is proved.
[problem 1] Use Rolle’s theorem to show that the function has a critical number in the interval . Then find the value of this critical number.
The critical number is

For the next example we need to recall that a root of a polynomial, , is a value , such that . We also need the fact that a quadratic polynomial has at most two roots, since if then by the quadratic formula

[problem 2a] Show that the polynomial has exactly one root.
Use the Intermediate Value Theorem to show that there is a root in the interval
Note that the derivative, , is strictly positive
Use Rolle’s Theorem to conclude that cannot have two roots
Argue by way of contradiction- suppose has two roots
[problem 2b] Use Rolle’s Theorem and the fact that a polynomial of degree three has at most three roots to prove that a polynomial of degree four has at most four roots.
Use an argument similar to example 2

2 The Mean Value Theorem

The Mean Value Theorem is one of the most far-reaching theorems in calculus. It states that for a continuous and differentiable function, the average rate of change over an interval is attained as an instantaneous rate of change at some point inside the interval. The precise mathematical statement is as follows.

The Mean Value Theorem has both geometric and conceptual interpretations. Geometrically, the equation asserts that the tangent line at and the secant line on are parallel, as shown in the figure below. Conceptually, the equation tells us that at the point , the instantaneous rate of change of is equal to the average rate of change of over the interval .

Conceptually, the left-hand side represents the instantaneous rate of change of at , while the right-hand side represents the average rate of change of over the interval . As an example of the conceptual interpretation of the theorem, consider a car that averages a speed of, say, 47.4 miles per hour on a long trip. The Mean Value Theorem tells us that the car must have been traveling at exactly 47.4 miles per hour at some time during the trip.

The Mean Value Theorem is an existence theorem because it asserts that there exists at least one value inside the interval that satisfies the equation In the examples and problems below, we will determine this special value.

(problem 3a) Is the function continuous on the closed interval and differentiable on the open interval ? Why? If yes to both, find all values that satisfy the conclusion of the Mean Value Theorem for on the interval .
Compute and
Solve the equation: for

The value of is: What do you notice about this value of in relation to the given interval? Is this always the case?

(problem 3b) Is the function continuous on the closed interval and differentiable on the open interval ? Why? If yes to both, find all values that satisfy the conclusion of the Mean Value Theorem for on the interval .
Compute and
Solve the equation: for

The value of is: What do you notice about this value of in relation to the given interval? Is this always the case?

(problem 3c) Is the function continuous on the closed interval and differentiable on the open interval ? Why? If yes to both, find all values that satisfy the conclusion of the Mean Value Theorem for on the interval .
Compute and
Solve the equation: for

The value of is: What do you notice about this value of in relation to the given interval? Is this always the case?

(problem 3d) Is the function continuous on the closed interval and differentiable on the open interval ? Why? If yes to both, find all values that satisfy the conclusion of the Mean Value Theorem for on the interval .
Compute and
Solve the equation: for

The values of in ascending order are: and are these values in the interval ?

(problem 4a) Is the function continuous on the closed interval and differentiable on the open interval ? Why? If yes to both, find all values that satisfy the conclusion of the Mean Value Theorem for on the interval .
Compute and
Solve

The value of is: Is this value in the interval ?

(problem 4b) Is the function continuous on the closed interval and differentiable on the open interval ? Why? If yes to both, find all values that satisfy the conclusion of the Mean Value Theorem for on the interval .
Compute and
Solve

The value of is: Is this value in the interval ?

(problem 4c) Is the function continuous on the closed interval and differentiable on the open interval ? Why? If yes to both, find all values that satisfy the conclusion of the Mean Value Theorem for on the interval .

Given that the function is continuous on the closed interval and differentiable on the open interval , find the values of which satisfy the conclusion of the Mean Value Theorem for on the interval .

Compute and
Solve

The value of is: Is this value in the interval ?

(problem 5) Is the function continuous on the closed interval and differentiable on the open interval ? Why? If yes to both, find all values that satisfy the conclusion of the Mean Value Theorem for on the interval .
Compute both and
Solve the equation for

The value of is: Is this value in the interval ?

(problem 6a) Is the function continuous on the closed interval and differentiable on the open interval ? Why? If yes to both, find all values that satisfy the conclusion of the Mean Value Theorem for on the interval .
Compute and
and
Solve the equation for
Begin by taking the reciprocal of both sides

The value of is: Is this value in the interval ?

(problem 6b) Is the function continuous on the closed interval and differentiable on the open interval ? Why? If yes to both, find all values that satisfy the conclusion of the Mean Value Theorem for on the interval .
Compute and
Solve the equation for
Only consider solutions in the interval

The values of are (in ascending order): , and Are these values in the interval ?

(problem 6c) Determine if the function continuous on the closed interval and differentiable on the open interval . Justify your reasoning for both continuity and differentiability. Additionally, assess whether there are any values of that satisfy the conclusion of the Mean Value Theorem for on the interval . Provide supporting explanations for your findings.

Is the function continuous on the closed interval and differentiable on the open interval ? Why or why not? Determine if there are any values of that satisfy the conclusion of the Mean Value Theorem for on the interval .

Compute and
Solve the equation for if possible

Were there any values of ? Does this contradict the Mean Value Theorem?

Here is a detailed, lecture style video on the Mean Value Theorem:
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2024-11-04 13:15:46