# MVT/Rolle's Theorem: Defining Intervals

• pyrosilver
In summary, the conversation is about defining the intervals where the Mean Value Theorem (MVT) and Rolle's theorem apply for a given function. It is mentioned that for MVT, the function must be continuous on the interval and differentiable on the open interval. It is also stated that Rolle's theorem is defined as f(a) = f(b), not f(a) = (f(b) = 0. It is then discussed that for MVT, any interval that does not include the critical point (in this case, x=4) is valid, while for Rolle's theorem, there is no valid interval as the function is never equal to 0 and therefore does not have a critical point.
pyrosilver

## Homework Statement

Define interval where a) MVT applies, b) Rolle's applies, c) MVT doesn't apply. explain. (In this, we're saying Rolle's is f(a) = f(b), not f(a) = (f(b) = 0.

## The Attempt at a Solution

f(x) = (x-4)^(2/3)

a) for MVT, must be cont on [a,b] and diff on (a,b). It is not diff at x=4, but for interval [5,8] it should be fine, so a = [5,8]?
b) I'm confused for this one. f(1) = 2.08. f(7) = 2.08. However, x is not diff at 4, so [1,7] isn't a valid interval, right? What is a valid interval, one that let's Rolle's apply?
c) MVT doesn't apply on [1,7] because x is not diff at 4? Is that okay?

pyrosilver said:

## Homework Statement

Define interval where a) MVT applies, b) Rolle's applies, c) MVT doesn't apply. explain. (In this, we're saying Rolle's is f(a) = f(b), not f(a) = (f(b) = 0.

## The Attempt at a Solution

f(x) = (x-4)^(2/3)

a) for MVT, must be cont on [a,b] and diff on (a,b). It is not diff at x=4, but for interval [5,8] it should be fine, so a = [5,8]?
Yes, any interval that does not include 4 works.

b) I'm confused for this one. f(1) = 2.08. f(7) = 2.08. However, x is not diff at 4, so [1,7] isn't a valid interval, right? What is a valid interval, one that let's Rolle's apply?
You are correct that [1, 7] does not work. You want to find some a, b, such that a and b are both less than 4 or both larger than 4 such that f(a)= f(b).
It is easy to see that f(x) is increasing for x> 4 and decreasing for x< 4. That is, in order that f(a)= f(b), either a= b or one is less than 4 and the other larger than 4. Neither gives a valid interval. Notice that if there were a valid interval, Rolle's theorem would say that f'(c)= 0 at some point inside that interval. And f' is NEVER 0 for this function. There is no such interval.

c) MVT doesn't apply on [1,7] because x is not diff at 4? Is that okay?
Yes, or any interval including 4.

## What is the MVT/Rolle's Theorem?

The Mean Value Theorem (MVT) and Rolle's Theorem are important theorems in calculus that are used to prove the existence of critical points and intervals on a given function. They both involve the concept of differentiation and help in understanding the behavior of a function.

## What is the significance of defining intervals in MVT/Rolle's Theorem?

Defining intervals is important in MVT/Rolle's Theorem because it helps in proving the existence of critical points or intervals on a given function. It also helps in understanding the behavior of a function and finding its maximum and minimum values.

## How is MVT/Rolle's Theorem used in real-world applications?

MVT/Rolle's Theorem is used in various real-world applications such as optimization problems in economics, physics, and engineering. It is also used in analyzing the speed and acceleration of moving objects and in finding the maximum and minimum values of a function in business and finance.

## What are the conditions for applying MVT/Rolle's Theorem?

The conditions for applying MVT/Rolle's Theorem are that the function must be continuous on a closed interval, and differentiable on the open interval within that closed interval. Additionally, the function must have equal values at the endpoints of the interval, and the derivative of the function must be equal to zero at some point within the interval.

## How is MVT/Rolle's Theorem related to the concept of critical points?

MVT/Rolle's Theorem is closely related to the concept of critical points. Critical points are points where the derivative of a function is equal to zero, and they play a crucial role in determining the behavior of a function. MVT/Rolle's Theorem helps in proving the existence of critical points and identifying the intervals in which they occur on a given function.

Replies
2
Views
2K
Replies
1
Views
2K
Replies
1
Views
2K
Replies
3
Views
834
Replies
2
Views
4K
Replies
16
Views
2K
Replies
4
Views
2K
Replies
26
Views
2K
Replies
4
Views
2K
Replies
1
Views
1K