AP Calculus 5.1 Rolle's Theorem

Which of the following is NOT a condition for Rolle's Theorem to apply to a function f on a closed interval [a, b]?

• f is continuous on [a, b]

• f is differentiable on (a, b)

• f(a) = f(b)

• There is at least one number c in (a, b) for which f'(c) = 0

If a function has a sharp point (cusp) within the open interval (a, b), which condition of Rolle's Theorem is violated?

• The function is continuous on the closed interval [a, b]

• The function is differentiable on the open interval (a, b)

• The function values at the endpoints are equal, f(a) = f(b)

• The function is defined on the closed interval [a, b]

For the function f(x) = x^3 - 6x^2 on the interval [0, 6], what is true about its continuity and differentiability?

• It is continuous but not differentiable

• It is both continuous and differentiable

• It is differentiable but not continuous

• It is neither continuous nor differentiable

The graph of the derivative of f(x) = x^3 - 6x^2 shows that f'(x) = 0 at which point within the open interval (0, 6)?

• x = 0

• x = 2

• x = 4

• x = 6

For the function f(x) = x^2 - 5x + 6, what value of c is guaranteed by Rolle's Theorem in the open interval formed by its x-intercepts?

• 2

• 2.5

• 3

• 5