# Does a Function Exist Given Specific Function Values and Derivative Constraints?

• Loppyfoot
In summary, the mean value theorem can be used to determine the existence of a function f when given specific conditions, such as f(0)=-1, f(2)=4, and f'(x)≤2 for all values of x. In this case, the mean value theorem tells us that there exists a number c in the interval (0,2) where f'(c) is equal to the average rate of change between f(0) and f(2). However, since this average rate of change is greater than the given limit of 2, the function would not exist.
Loppyfoot

## Homework Statement

Does the function f exist when f(0)=-1, f(2)=4, and f'(x)≤2 for all values of x? Justify your answer.

## The Attempt at a Solution

I am having some trouble with where to start...and end. Thanks for your help.

What would the mean value theorem tell you about f(x) on the interval [0,2]?

What would the mean value theorem tell you about f(x) on the interval [0,2]?
It tells me that there exists a number c in (0,2) such that f'(c) = (f(2)-f(0))/(2-0)

Ok. What is (f(2)-f(0))/(2-0) as a number?

so 2.5 is not less than or equal to 2, so the function would not exist?

Loppyfoot said:
so 2.5 is not less than or equal to 2, so the function would not exist?

Exactly.

Great. Thank you, Sir.

## 1. What is Rolle's/MVT Problem?

Rolle's/MVT Problem refers to two related theorems in calculus - Rolle's Theorem and the Mean Value Theorem. These theorems are used to analyze the behavior of a function on an interval and provide important insights into its properties.

## 2. What is Rolle's Theorem?

Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, and the function's values at the endpoints of the interval are equal, then there exists at least one point within the interval where the derivative of the function is equal to 0.

## 3. What is the Mean Value Theorem?

The Mean Value Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point within the interval where the slope of the tangent line to the function is equal to the slope of the secant line connecting the endpoints of the interval.

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

Rolle's Theorem is a special case of the Mean Value Theorem, where the slope of the secant line is equal to 0. In other words, Rolle's Theorem can be seen as a corollary of the Mean Value Theorem.

## 5. How are Rolle's/MVT Problem used in real-life applications?

Rolle's/MVT Problem can be used to analyze the behavior of real-world phenomena that can be modeled using functions, such as speed and acceleration of a moving object or the rate of change in a chemical reaction. These theorems provide important insights into the behavior of these phenomena and can help in making predictions or optimizing certain processes.

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