The Theorems of Calculus
The Intermediate Value Theorem If a function f is continuous on the closed interval [a, b]
and f(a) ≠ f(b), then for any k , between f(a) and f(b) there exists a c between a and b such that
f(c) = k.
The Extreme Value Theorem If a function f is continuous on the closed interval [a, b] , then f
has both a maximum value and a minimum value on the interval.
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 number c in (a, b) such
f (b) − f (a)
that f ′(c) =
.
b−a
Rolle’s Theorem If a function f is continuous on the closed interval [a, b] and differentiable on
the open interval (a, b) and f (a) = f (b) = 0, then there exists at least one number c in (a, b) such
that f ′(c) = 0.
Absolute Extremum Theorem If a function f is continuous on an interval and has only one
relative extremum, f(c), on the interval, then f(c) is also an absolute extremum on the interval.
Differentiability ⇒ Continuity If a function f is differentiable at x = a, then f is continuous
at x = a.
1. (AB90-7) A function f that is continuous for all real numbers x has f (3) = -1 and f (7) = 1. If f (x)
= 0 for exactly one value of x, then which of the following could be x?
a) -1
b) 0
c) 1
d) 4
e) 9
2. (BC73-18) Let g be a continuous function on the closed interval [0, 1]. Let g(0) = 1 and g(1) = 0.
Which of the following is NOT necessarily true?
a) There exists a number h in [0,1] such that g(h) ≥ g(x) for all x in [0,1].
b) For all a and b in [0,1], if a = b, then g(a) = g(b).
1
.
2
3
d) There exists a number h in [0, 1] such that g(h) = .
2
c) There exists a number h in [0, 1] such that g(h) =
e) For all h in the open interval (0, 1), lim g ( x) = g (h) .
x →h
3. (AB98-4) If f is continuous for a ≤ x ≤ b and differentiable for a < x < b, which of the following
could be false?
f (b) − f (a)
for some c such that a < c < b.
b−a
b) f ′(c) = 0 for some c such that a < c < b.
c) f has a minimum value on a ≤ x ≤ b .
d) f has a maximum value on a ≤ x ≤ b .
a) f ′(c) =
b
e)
∫ f ( x)dx exists.
a
4. (AB90-23 calc) Let f be the function f ( x) = cos( x) . If the number c satisfies the conclusion of the
Mean Value Theorem for f on the closed interval [2, 4], then c is
a) 0.119
b) 2.902
c) 3.023
d) 3.142
e) 3.261
π
3π
⎛ x⎞
that satisfies
⎟ , then there exists a number c in the interval < x <
2
2
⎝2⎠
5. (AB93-18) If f(x)= sin ⎜
the conclusion of the Mean Value Theorem. Which of the following could be c ?
a)
2π
3
b)
3π
4
c)
5π
6
d) π
e)
3π
2
6. (AB88-20) Let f be a polynomial function with degree greater than 2. If a ≠ b , and
f(a) = f(b) = 1, which of the following must be true for at least one value of x between a and b?
I.
II.
III.
a) none
f(x) = 0
f ′( x) = 0
f ′′( x) = 0
b) I only
x
f(x)
c) II only
d) I and II only e) I, II, and III
0
1
1
k
2
2
7. (AB98-26) The function f is continuous on the closed interval [0, 2] and has values that are given in
the table above. The equation f ( x) =
a) 0
b)
1
2
1
must have at least two solutions in the interval [0, 2] if k =
2
c) 1
d) 2
e) 3
8. (AB98-91) Let f be a function that is differentiable on the open interval (1, 10). If f(2) = -5, f(5) = 5
and f(9) = -5, which of the following must be true?
I. f has at least two zeros.
II. The graph of f has at least one horizontal tangent.
III. For some c, 2 < c < 5, f (c) = 3.
a) none
Solutions 1) d
b) I only
2) d
3) b
c) I and II only d) I and III only
4) c
5) d
6) c
7) a
8) e
e) I, II, and III