Not checking the conditions

Oct 13, 2025 by windailys.com 64 views

Which of the Following Functions Does Not Satisfy the Conditions of the Mean Value Theorem on [3,8]?

The Mean Value Theorem (MVT) is a fundamental concept in calculus that provides a relationship between the derivative of a function and the difference of its values at two points. It states that if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a). In this article, we will examine four functions to determine which one does not satisfy the conditions of the Mean Value Theorem on the interval [3, 8].

Understanding the Conditions of the Mean Value Theorem

Before we proceed to analyze the functions, let's recall the conditions of the Mean Value Theorem:

Continuity: The function f(x) must be continuous on the closed interval [a, b].
Differentiability: The function f(x) must be differentiable on the open interval (a, b).
Existence of a Point c: There must exist a point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Analyzing the Functions

Now, let's analyze each of the four functions to determine which one does not satisfy the conditions of the Mean Value Theorem on the interval [3, 8].

A. $f(x)=|x+5|$

The function f(x) = |x + 5| is a piecewise function defined as:

f(x) = x + 5, if x ≥ -5 f(x) = -(x + 5), if x < -5

On the interval [3, 8], the function is defined as f(x) = x + 5. This function is continuous and differentiable on the interval [3, 8]. However, the derivative of the function is not continuous at x = -5, which is not in the interval [3, 8]. Therefore, the function f(x) = |x + 5| does not satisfy the conditions of the Mean Value Theorem on the interval [3, 8].

B. $f(x)=|x-5|$

The function f(x) = |x - 5| is a piecewise function defined as:

f(x) = x - 5, if x ≥ 5 f(x) = -(x - 5), if x < 5

On the interval [3, 8], the function is defined as f(x) = -(x - 5). This function is continuous and differentiable on the interval [3, 8]. The derivative of the function is f'(x) = -1, which is continuous on the interval [3, 8]. Therefore, the function f(x) = |x - 5| satisfies the conditions of the Mean Value Theorem on the interval [3, 8].

C. $f(x)=\sqrt{x+5}$

The function f(x) = √(x + 5) is defined as:

f(x) = √(x + 5), if x ≥ -5

On the interval [3, 8], the function is defined as f(x) = √(x + 5). This function is continuous and differentiable on the interval [3, 8]. The derivative of the function is f'(x) = 1 / (2√(x + 5)), which is continuous on the interval [3, 8]. Therefore, the function f(x) = √(x + 5) satisfies the conditions of the Mean Value Theorem on the interval [3, 8].

D. $f(x)=x^2-5x$

The function f(x) = x^2 - 5x is a quadratic function defined as:

f(x) = x^2 - 5x

On the interval [3, 8], the function is continuous and differentiable. The derivative of the function is f'(x) = 2x - 5, which is continuous on the interval [3, 8]. Therefore, the function f(x) = x^2 - 5x satisfies the conditions of the Mean Value Theorem on the interval [3, 8].

Conclusion

In conclusion, the function f(x) = |x + 5| does not satisfy the conditions of the Mean Value Theorem on the interval [3, 8] because it is not differentiable on the open interval (3, 8). The other three functions, f(x) = |x - 5|, f(x) = √(x + 5), and f(x) = x^2 - 5x, satisfy the conditions of the Mean Value Theorem on the interval [3, 8].

References

Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
Anton, H. (2016). Calculus: A New Horizon. John Wiley & Sons.
Thomas, G. B. (2016). Calculus and Analytic Geometry. Addison-Wesley.

Keywords: Mean Value Theorem, Continuity, Differentiability, Piecewise Functions, Quadratic Functions.
Frequently Asked Questions (FAQs) About the Mean Value Theorem

Q: What is the Mean Value Theorem?

A: The Mean Value Theorem states that if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Q: What are the conditions of the Mean Value Theorem?

A: The conditions of the Mean Value Theorem are:

Continuity: The function f(x) must be continuous on the closed interval [a, b].
Differentiability: The function f(x) must be differentiable on the open interval (a, b).
Existence of a Point c: There must exist a point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Q: What is the significance of the Mean Value Theorem?

A: The Mean Value Theorem is significant because it provides a relationship between the derivative of a function and the difference of its values at two points. This relationship is useful in many applications, such as optimization problems, physics, and engineering.

Q: How do I apply the Mean Value Theorem?

A: To apply the Mean Value Theorem, follow these steps:

Check the conditions: Check if the function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b).
Find the derivative: Find the derivative of the function f(x).
Find the point c: Find the point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Q: What are some common mistakes to avoid when applying the Mean Value Theorem?

A: Some common mistakes to avoid when applying the Mean Value Theorem are:

Not checking the conditions: Make sure to check if the function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b).
Not finding the derivative: Make sure to find the derivative of the function f(x).
Not finding the point c: Make sure to find the point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Q: Can the Mean Value Theorem be applied to all functions?

A: No, the Mean Value Theorem cannot be applied to all functions. The function must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b).

Q: What are some real-world applications of the Mean Value Theorem?

A: Some real-world applications of the Mean Value Theorem include:

Optimization problems: The Mean Value Theorem is used to solve optimization problems, such as finding the maximum or minimum of a function.
Physics: The Mean Value Theorem is used to describe the motion of objects, such as the velocity and acceleration of a particle.
Engineering: The Mean Value Theorem is used to design and optimize systems, such as electronic circuits and mechanical systems.

Conclusion

In conclusion, the Mean Value Theorem is a fundamental concept in calculus that provides a relationship between the derivative of a function and the difference of its values at two points. By understanding the conditions of the Mean Value Theorem and how to apply it, you can solve a wide range of problems in mathematics, physics, and engineering.

References

Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
Anton, H. (2016). Calculus: A New Horizon. John Wiley & Sons.
Thomas, G. B. (2016). Calculus and Analytic Geometry. Addison-Wesley.

Keywords: Mean Value Theorem, Continuity, Differentiability, Optimization Problems, Physics, Engineering.