Understanding Rolle's Theorem

The video tutorial explains Rolle's Theorem, which states that if a function is continuous on a closed interval, differentiable on an open interval, and the function values at the endpoints are equal, then there exists at least one point where the derivative is zero. The video provides multiple examples, including polynomial, sine, and square root functions, to illustrate the theorem's application. It also discusses cases where the theorem does not apply, such as non-differentiable and absolute value functions.

What is the main conclusion of Rolle's Theorem if all conditions are met?

• The function is not differentiable.

• The function has a vertical tangent.

• There exists a point where the derivative is zero.

• The function is continuous everywhere.

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

• The function's values at the endpoints must be equal.

• The function must be continuous on a closed interval.

• The function must be differentiable on an open interval.

• The function must have a maximum value.

In a graph with a sharp turn, why can't Rolle's Theorem be applied?

• The function is not continuous.

• The function is not differentiable.

• The function has equal endpoint values.

• The function has a horizontal tangent.

For the function f(x) = x^2 - 5x + 3 on the interval [0, 5], what is the value of C where f'(C) = 0?

• 4.5

• 3.5

• 2.5

• 1.5

Why can Rolle's Theorem be applied to the function f(x) = sin(x) on the interval [0, 2π]?

• The function has a vertical tangent.

• The function is not continuous.

• The function is not differentiable.

• The function has equal values at the endpoints.

What are the values of C for f(x) = sin(x) on [0, 2π] where f'(C) = 0?

• π/2 and 3π/2

• π and 2π

• 0 and π

• π/4 and 3π/4

Why can't Rolle's Theorem be applied to f(x) = x^(2/3) + 1 on [-4, 4]?

• The function has a cusp.

• The function has equal endpoint values.

• The function is differentiable.

• The function is continuous.

For the function f(x) = x√(4-x) on [0, 4], what is the approximate value of C where f'(C) = 0?

• 4.67

• 3.67

• 2.67

• 1.67

What is the main reason Rolle's Theorem cannot be applied to an absolute value function like |x-1| on [-1, 3]?

• The function is continuous.

• The function has a sharp turn.

• The function is differentiable.

• The function has equal endpoint values.

Which of the following functions is most likely to have a horizontal tangent within a given interval?

• A function with a vertical asymptote.

• An absolute value function.

• A polynomial function.

• A function with a cusp.