Mean Value Theorem Practice
1. a. Restate the Mean Value Theorem in your own words. Then, create a graph to show the
geometric interpretation of the theorem.
b. Restate Rolle’s Theorem in your own words. Then, create a graph to show the geometric
interpretation of this theorem.
2. Here is the first Free Response Question from the AP test given in 1989. (Non-Calculator)
Let f be the function given by f(x) = x3 – 7x + 6
a. Find the zeros of f
b. Write an equation of the line tangent to the graph
of f at x = -1
c. Find the number c that satisfies the conclusion of the Mean Value Theorem for f on the closed
interval [1,3].
3. Consider the curve y2 = 4 + x and chord AB joining the points A(-4,0) and B(0,2) on the curve.
Find the x- and y-coordinates of the point on the curve where the tangent line is parallel to chord
AB. (Non-Calculator)
_____4. Consider the graph of f(x) = xsin(x) on the domain [-4,4]. How many values of c in (-4,4)
appear to satisfy the Mean Value Theorem?
Is this also an example of Rolle’s
theorem? Explain your answer.
A. None
B. One
C. Two
D. Three
E. Four or More
_____5. Find a value c, for x, that satisfies the conclusion of the Mean Value Theorem for
Derivatives for f(x) = 3x2 – 5x + 1 on the interval [2,5].
A. 1
D.
23
6
B.
13
6
E.
7
2
C.
11
6
_____6. Find the value of c that satisfies the Mean Value Theorem for derivatives on the
interval [0,5] for the function f(x) = x3 – 6x is
A. 
D.
5
3
5
3
B. 0
E.
C. 1
5
3
_____7. If f is a continuous function on the closed interval [a,b], which of the following must be
true?
A. There is a number c in the open interval (a,b), such that f(c) = 0.
B. There is a number c in the open interval (a,b) such that f(a) < f(c) < f(b)
C. There is a number c in the closed interval [a,b] such that f(c) < f(x) for all x in [a,b]
D. There is a number c in the open interval (a,b) such that f’(c) = 0
E. There is a number c in the open interval (a,b) such that f'(c) 
f(b)  f(a)
ba