Understanding Rolle's Theorem

The video tutorial explains how to apply Rolle's Theorem to a given function on a closed interval. It begins by introducing the function and the theorem, then provides a detailed explanation of the theorem's conditions. The tutorial proceeds to apply the theorem to the function, verifying the necessary conditions and calculating the derivative. Finally, it solves for the value of C where the derivative equals zero, confirming the solution graphically.

What is the function given in the problem?

• f(x) = x^(1/2) - x

• f(x) = x + x^(1/2)

• f(x) = x - x^(1/2)

• f(x) = x^2 - x

Which of the following is NOT a condition for Rolle's Theorem?

• The function values at the endpoints must be equal.

• The function must be differentiable on an open interval.

• The function values at the endpoints must be different.

• The function must be continuous on a closed interval.

What does Rolle's Theorem guarantee if its conditions are met?

• The function has a maximum at the midpoint.

• The function is increasing throughout the interval.

• There is at least one point where the derivative is zero.

• The function is decreasing throughout the interval.

What are the values of f(0) and f(1) for the given function?

• f(0) = 0, f(1) = 1

• f(0) = 1, f(1) = 0

• f(0) = 1, f(1) = 1

• f(0) = 0, f(1) = 0

What is the graphical significance of the point where the derivative is zero?

• The function has a vertical tangent line.

• The function has a horizontal tangent line.

• The function has a discontinuity.

• The function has a point of inflection.

How is the derivative of the function expressed?

• f'(x) = 1 + 1/2x^(1/2)

• f'(x) = 1 - 1/2x^(1/2)

• f'(x) = 1 - 1/2x^(-1/2)

• f'(x) = 1 + 1/2x^(-1/2)

What is the value of c where the derivative is zero?

• c = 1/2

• c = 1/3

• c = 1/5

• c = 1/4

Why is it important to verify that c = 1/4 is within the interval?

• To validate the application of Rolle's Theorem.

• To check if the function has a maximum.

• To confirm the function is differentiable.

• To ensure the function is continuous.

What does the horizontal tangent line at x = 1/4 indicate?

• The function is increasing at this point.

• The function is decreasing at this point.

• The slope of the tangent line is zero.

• The function has a discontinuity.

What is the final conclusion about the function based on Rolle's Theorem?

• The function has no points where the derivative is zero.

• The function has multiple points where the derivative is zero.

• The function has exactly one point where the derivative is zero.

• The function is not differentiable.