PrepGo

AP Calculus BC Practice Quiz: Working with the Intermediate Value Theorem (IVT)

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 7 questions to check your progress.

Question 1 of 7

Let f be a continuous function on the closed interval [2, 8]. If f(2) = 10 and f(8) = 30, which of the following values is guaranteed to exist for some c in (2, 8) by the Intermediate Value Theorem?

All Questions (7)

Let f be a continuous function on the closed interval [2, 8]. If f(2) = 10 and f(8) = 30, which of the following values is guaranteed to exist for some c in (2, 8) by the Intermediate Value Theorem?

A) f(c) = 5

B) f(c) = 25

C) f(c) = 35

D) f(c) = 10

Correct Answer: B

The Intermediate Value Theorem guarantees that for any value d between f(a) and f(b), there is a c such that f(c) = d. Here, f(2) = 10 and f(8) = 30. The only option between 10 and 30 is 25.

According to the formal definition, which of the following is a necessary condition for the Intermediate Value Theorem to apply to a function f on a closed interval [a, b]?

A) The function f must be increasing on [a, b].

B) The function f must be differentiable on [a, b].

C) The function f must be continuous on [a, b].

D) The function f must have a positive value at f(a) or f(b).

Correct Answer: C

The provided text explicitly states that the Intermediate Value Theorem applies 'If f is a continuous function on the closed interval [a, b]'. Continuity is a required condition.

Let f be a continuous function on the closed interval [a, b]. If f(a) is negative and f(b) is positive, what behavior does the Intermediate Value Theorem guarantee on the interval?

A) The function has a minimum value at x=a.

B) There is exactly one number c between a and b such that f(c) = 0.

C) The function is always increasing on the interval [a, b].

D) There is at least one number c between a and b such that f(c) = 0.

Correct Answer: D

Since f is continuous and 0 is a number between f(a) (a negative value) and f(b) (a positive value), the IVT guarantees there is at least one number c between a and b such that f(c) = 0. This explains the behavior of the function by guaranteeing it crosses the x-axis.

A function f is continuous on the closed interval [-1, 5]. Given f(-1) = -4 and f(5) = 12, the Intermediate Value Theorem guarantees that for any d in which of the following intervals, there is a c in (-1, 5) such that f(c) = d?

A) [-1, 5]

B) (-∞, ∞)

C) [12, -4]

D) [-4, 12]

Correct Answer: D

The Intermediate Value Theorem applies to any value d that is between f(a) and f(b). In this case, f(a) = -4 and f(b) = 12. The interval containing all numbers between -4 and 12 is [-4, 12].

If the conditions for the Intermediate Value Theorem are met for a function f on an interval [a, b], and d is a value between f(a) and f(b), what precise conclusion can be drawn about the number of values c such that f(c) = d?

A) There is exactly one such c.

B) There is at least one such c.

C) There are at most two such c values.

D) There are no such c values.

Correct Answer: B

The theorem states that 'there is at least one number c'. It does not specify an upper limit; there could be one, two, or more such values, but it only guarantees a minimum of one.

Let f be a continuous function on [0, 10] with f(0) = -5 and f(10) = 15. Which of the following statements is NOT guaranteed by the Intermediate Value Theorem?

A) There exists a c in (0, 10) such that f(c) = 10.

B) There exists a c in (0, 10) such that f(c) = 0.

C) There exists a c in (0, 10) such that f(c) = 20.

D) The function's behavior can be partially explained by showing it must take on all values between -5 and 15.

Correct Answer: C

The Intermediate Value Theorem only guarantees the existence of f(c) = d for values of d *between* f(a) and f(b). Since f(0) = -5 and f(10) = 15, the theorem applies to any d in [-5, 15]. The value 20 is outside this range, so its existence is not guaranteed by the IVT.

The function h(x) is continuous on the interval [1, 4]. Selected values of h(x) are given in the table below. | x | 1 | 2 | 3 | 4 | |---|---|---|---|---| | h(x)| 10| -2| 1 | 5 | Based on the Intermediate Value Theorem, on which interval must there be a value c such that h(c) = 0?

A) [2, 3]

B) [3, 4]

C) [1, 2]

D) The theorem does not guarantee a value of 0 on any of these subintervals.

Correct Answer: C

To guarantee h(c) = 0, we need to find an interval [a, b] where 0 is between h(a) and h(b). On the interval [1, 2], the function values go from h(1) = 10 to h(2) = -2. Since 0 is between 10 and -2, the IVT guarantees a c in (1, 2) such that h(c) = 0. On [2, 3], values go from -2 to 1, so a root is also guaranteed there. However, [1, 2] is the first option that satisfies the condition.