1) When are Rolle's theorem and the Mean Value Theorem equivalent?

1. f(a) > f(b)
2. f(a) = f(b)
3. f(a) f(b)
4. f(a) = f(b)
5. f(a) < f(b)

2) Graph the function on a calculator and draw the secant line that connects the endpoints. Estimate the number of points c such that f'(c)(b a) = f(b) f(a).

y = 6x³ + 4x + 2 over [1, 1]

3) Use the Mean Value Theorem and find all points 0 < c < 4

such that f(4) f(0) = f'(c)(4 0).

(Enter your answers as a comma-separated list.)

f(x) = x³

c =

4) Determine whether the Mean Value Theorem applies for the function over the given interval [a, b].

Justify your answer. (Select all that apply.)

y = x³ + 3x + 2 over [0, 8]

1. The Mean Value Theorem does apply.
2. The Mean Value Theorem does not apply since the function is discontinuous at a point.
3. The Mean Value Theorem does not apply since the function is not differentiable at a point.
4. The Mean Value Theorem does not apply since the f(a) does not exist.
5. The Mean Value Theorem does not apply since the f(b) does not exist.