# Mean Value Theorem/Rolle's Theorem and differentiability

• NanaToru
In summary: It gets worse. Wait until you've been coding for 30 years, you're teaching a class, you type in the code that you're absolutely sure will give the right answer and the computer says "What the Hell is that?!?!?!"In summary, the conversation discusses the function f(x) = 1 - x2/3 and its properties, including the fact that f(-1) = f(1) and there is no number c in the interval (-1,1) such that f'(c) = 0. This is not a contradiction of Rolle's Theorem because the function does not satisfy all the hypotheses of the theorem, namely being differentiable on the interval. The conversation also touches on the concept
NanaToru

## Homework Statement

Let f(x) = 1 - x2/3. Show that f(-1) = f(1) but there is no number c in (-1,1) such that f'(c) = 0. Why does this not contradict Rolle's Theorem?

## The Attempt at a Solution

f(x) = 1 - x2/3.
f(-1) = 1 - 1 = 0
f(1) = 1 - 1 = 0

f' = 2/3 x -1/3.

I don't understand why this doesn't have a number c in f'(c), or why Rolle's theorem excludes nondifferentiable points?

NanaToru said:

## Homework Statement

Let f(x) = 1 - x2/3. Show that f(-1) = f(1) but there is no number c in (-1,1) such that f'(c) = 0. Why does this not contradict Rolle's Theorem?

## The Attempt at a Solution

f(x) = 1 - x2/3.
f(-1) = 1 - 1 = 0
f(1) = 1 - 1 = 0

f' = 2/3 x -1/3.

I don't understand why this doesn't have a number c in f'(c), or why Rolle's theorem excludes nondifferentiable points?
For a function to violate Rolle's theorem, it would need to do two things:
1. Satisfy all the hypotheses of Rolle's theorem.
2. Fail to satisfy the conclusion of Rolle's theorem.

Does this function do those two things?

NanaToru
Hm... I'm not sure it satisfies all the hypotheses--from what the back of the book says, it isn't differentiate on the interval of (-1, 1) but I'm not sure how? Did I do a bad job differentiating it?

What do you get for f'(0)?

Is it not 0? Or is that not a valid answer?

NanaToru said:
Is it not 0? Or is that not a valid answer?
If you think ##\frac 1 0 = 0## then you must also think ##0\cdot 0 = 1##?

...This is honestly the most embarrassing moment of my life. Thank you though!

NanaToru said:
...This is honestly the most embarrassing moment of my life. Thank you though!
Just wait!

WWGD

## 1. What is the Mean Value Theorem?

The Mean Value Theorem is a fundamental theorem in calculus that states that for any continuous and differentiable function on an interval, there exists at least one point where the slope of the tangent line is equal to the average rate of change of the function over that interval.

## 2. What is Rolle's Theorem?

Rolle's Theorem is a special case of the Mean Value Theorem where the function has the same value at the endpoints of the interval. It states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point where the derivative is equal to zero.

## 3. How are Mean Value Theorem and Rolle's Theorem related?

Rolle's Theorem is a special case of the Mean Value Theorem. Both theorems are used to prove the existence of a point where the derivative is equal to a certain value. However, Rolle's Theorem requires the function to have the same value at the endpoints of the interval, while the Mean Value Theorem only requires the function to be continuous and differentiable on the interval.

## 4. What is the significance of differentiability in these theorems?

Differentiability is a key requirement in both Mean Value Theorem and Rolle's Theorem. These theorems rely on the existence of the derivative of a function at a certain point, and without differentiability, the theorems cannot be applied. In other words, the functions must be smooth and have a well-defined tangent line at the point in question.

## 5. What are some real-world applications of Mean Value Theorem and Rolle's Theorem?

Mean Value Theorem and Rolle's Theorem are used to solve a variety of problems in fields such as physics, engineering, and economics. For example, they can be used to analyze the movement of objects, optimize production processes, and determine the maximum or minimum values of a function. They also have applications in the study of optimization and optimization algorithms.

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