Mean Value Theorem and Rolle’s Theorem
Rolle’s Theorem. Let f be a function satisfying all of the following three conditions:
1. f is continuous on the closed interval [a, b];
2. f is differentiable on the open interval (a, b);
3. f (a) = f (b).
Then there is at least one number c in (a, b) such that f � (c) = 0.
Mean Value Theorem (MVT). Let f be a function satisfying both of these conditions:
1. f is continuous on the closed interval [a, b];
2. f is differentiable on the open interval (a, b).
(a)
Then there is at least one number c in (a, b) such that f � (c) = f (b)−f
.
b−a
1. Illustrate Rolle’s Theorem graphically using the graph below of f on [a, b]. Illustrate
the Mean Value Theorem graphically using the graph below of g on [a, b]. Hint: use secant
and tangent lines.
2. Consider the function f (x) = x3 − 5x2 + 6x + 2 graphed below.
(a) Verify the three conditions of Rolle’s Theorem to explain why Rolle’s Theorem applies
to f on the interval [0, 3]. Then find the number(s) c such that f � (c) = 0. Illustrate your
result by marking and labeling the relevant points and tangent lines on the graph.
(b) As in part a, explain why the Mean Value Theorem applies to f on the interval [0, 4].
Find the number(s) that are guaranteed by the MVT. Illustrate your results by marking
points and drawing relevant secant and tangent lines.
3. The following exercises illustrate the importance of the conditions of the Mean Value
Theorem and why you must verify them in order to apply the conclusion of MVT.
(a) On the axes on the left, draw a function f on an interval [a, b] that is continuous on [a, b],
(a)
not differentiable on (a, b), and there is no c between a and b such that f � (c) = f (b)−f
.
b−a
(b) On the axes on the right, draw a function g on an interval [a, b] that is differentiable
on (a, b), not continuous on [a, b], and there is no c between a and b such that
(a)
g � (c) = f (b)−f
.
b−a
4. A consequence of the Mean Value Theorem is that if f � (x) > 0 for all x in (a, b) and f
is continuous on [a, b], then f (a) < f (b). Use the MVT to show this is true.