Math 400 (Section 4.6)
Exploration: Mean Value Theorem
Name ______________________________
You have recently proved Rolle's Theorem, which states that if f is continuous on the closed
interval [ a, b] and differentiable on the open interval ( a, b ) , and if f ( a ) = f ( b ) , then there is at
least one value c in the interval ( a, b ) such that f ′ ( c ) = 0 .
In this exploration, you will use Rolle's Theorem as a lemma to prove the Mean Value
Theorom, which is a generalization of Rolle's Theorem, and has very important implications for
what follows in calculus.
The Mean Value Theorem
If a function f is continuous on the closed interval [ a, b] and differentiable on the open interval
( a, b ) , then there exists at least one value c in the interval ( a, b ) such that
f (b ) − f ( a )
f ′ (c ) =
.
b−a
1. Explain why Rolle's Theorem is a special case of the Mean Value Theorem.
2. In a direct proof of a theorem, we assume the hypothesis of the theorem to be true, and then
show that the conclusion follows. Complete the sentence below, which assumes the
hypothesis of the MVT to be true.
Suppose that
3. Now let g ( x ) be the linear function that defines the secant line
through the points ( a, f ( a ) ) and ( b, f ( b ) ) as shown at right.
Write an equation for g ( x ) . [Hint: use point-slope form for the
equation of a line.]
4. Define h ( x ) by the equation h ( x ) = f ( x ) − g ( x ) . Does h ( x ) satisfy the conditions
(hypotheses) of Rolle's Theorem on the interval [ a, b] ? Discuss why or why not with your
group, and record your reasons here.
5. Since h ( x ) satisfies the hypotheses for Rolle's Theorem, the conclusion must follow.
Express the conclusion of Rolle's Theorem as it applies to h ( x ) .
6. Now use the definition of h ( x ) , along with the fact that the derivative of a difference of two
functions is the difference of derivatives of those two functions to reach the conclusion of the
Mean Value Theorem for f ( x ) on [ a, b] .
7. Now that you have proved the Mean Value Theorem, determine whether it applies to each of the
following functions on the given closed intervals. If so, find all c-values in the corresponding open
interval such that f ′ ( c ) =
f (b ) − f ( a )
.
b−a
(a) f ( x ) = x ( x 2 − x − 2 ) ; [ −1,1]
c = _________
(b) f ( x ) =
x +1
;
x
1 
 2 , 2 
(c) f ( x ) = 2sin x + sin 2 x; [ 0, π ] .
c = _________
c = _________