AP Calculus BC Practice Quiz: Determining Concavity of Functions over Their Domains

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

Test your understanding with short quizzes. This quiz has 10 questions to check your progress.

Question 1 of 10

Let f be a twice-differentiable function. If f''(x) > 0 for all x in the interval (a, b), which of the following statements must be true about the graph of f on that interval?

A)The graph of f is increasing.B)The graph of f is decreasing.C)The graph of f is concave up.D)The graph of f is concave down.

All Questions (10)

Let f be a twice-differentiable function. If f''(x) > 0 for all x in the interval (a, b), which of the following statements must be true about the graph of f on that interval?

A) The graph of f is increasing.

B) The graph of f is decreasing.

C) The graph of f is concave up.

D) The graph of f is concave down.

Correct Answer: C

According to the provided content, the second derivative of a function provides information about its concavity. Specifically, if the second derivative is positive on an interval, the graph of the original function is concave up on that interval.

The derivative of a function g is given by g'(x). If g'(x) is a decreasing function on the interval (0, 5), what can be concluded about the graph of g(x) on this interval?

A) The graph of g is concave up.

B) The graph of g is concave down.

C) The graph of g has a local maximum.

D) The graph of g has a point of inflection.

Correct Answer: B

The provided content states that the graph of a function is concave down on an open interval if the function’s derivative is decreasing on that interval. Since g'(x) is decreasing on (0, 5), the graph of g(x) must be concave down on (0, 5).

The graph of a twice-differentiable function h has a point of inflection at x = 2. Which of the following provides a sufficient justification for this conclusion?

A) h'(2) = 0

B) h''(2) = 0

C) The derivative h'(x) has a local extremum at x = 2.

D) The second derivative h''(x) changes sign at x = 2.

Correct Answer: D

A point of inflection occurs where the concavity of the function's graph changes. This change in concavity is indicated by a sign change in the second derivative, h''(x). While h''(2) is often zero at a point of inflection, this condition alone is not sufficient. The sign of h''(x) must change at x=2.

Let f be a function with a second derivative given by f''(x) = x^2(x - 3)(x + 1). On which of the following open intervals is the graph of f concave up?

A) (-∞, -1) only

B) (3, ∞) only

C) (-1, 0) U (3, ∞)

D) (-∞, -1) U (3, ∞)

Correct Answer: D

The graph of f is concave up where f''(x) > 0. We need to analyze the sign of f''(x) = x^2(x - 3)(x + 1). The critical points are x = -1, 0, and 3. Testing intervals: for x < -1, f'' is positive. For -1 < x < 0, f'' is negative. For 0 < x < 3, f'' is negative (the x^2 term is positive). For x > 3, f'' is positive. Therefore, f is concave up on (-∞, -1) and (3, ∞).

The graph of f', the derivative of a function f, is shown. For which of the following intervals is the graph of f concave down?

A) (-2, 1)

B) (1, 3)

C) (-1, 2)

D) (-2, 0)

Correct Answer: A

The graph of f is concave down on intervals where its derivative, f', is decreasing. Based on the provided graph of f', the function is decreasing on the interval (-2, 1). Therefore, the graph of f is concave down on this interval.

A function f is continuous on the closed interval [0, 4] and twice-differentiable on the open interval (0, 4). If f'(x) is an increasing function on (0, 4), which statement about the graph of f is correct?

A) The graph of f is concave down on (0, 4).

B) The graph of f is concave up on (0, 4).

C) The graph of f has a point of inflection in (0, 4).

D) The graph of f is a straight line.

Correct Answer: B

The provided content states that the graph of a function is concave up on an open interval if the function’s derivative is increasing on that interval. Since f'(x) is increasing on (0, 4), the graph of f(x) must be concave up on (0, 4).

Let g be a twice-differentiable function. The table below gives selected values of g''(x). Which of the following statements could be false? x | 0 | 1 | 2 | 3 | 4 ---|---|---|---|---|--- g''(x) | 5 | 2 | 0 | -1 | -3

A) The graph of g is concave up on the interval (0, 1).

B) The graph of g is concave down on the interval (3, 4).

C) The graph of g has a point of inflection at x = 2.

D) The derivative, g'(x), is decreasing on the interval (3, 4).

Correct Answer: C

A point of inflection requires the second derivative to change sign. At x=2, g''(x)=0, but we do not know if g''(x) changes sign exactly at x=2. It changes sign somewhere in the interval (1, 3), but not necessarily at x=2. Statement A is true because g''(x) > 0 on [0,1]. Statement B is true because g''(x) < 0 on [3,4]. Statement D is true because g is concave down on (3,4), which means g' must be decreasing.

The second derivative of a function f is given by f''(x). To justify that the graph of f is concave down on an interval, one must show that for all x in that interval:

A) f(x) < 0

B) f'(x) < 0

C) f'(x) is decreasing

D) f'(x) is increasing

Correct Answer: C

This question asks for the justification based on the first derivative. The provided content states that the graph of a function is concave down on an open interval if the function’s derivative is decreasing on that interval. This is a direct application of the definition relating the behavior of the first derivative to the concavity of the function.

A function f has a second derivative f''(x) = (x-a)^2. Which of the following statements is true about the graph of f at x=a?

A) The graph of f has a point of inflection at x=a.

B) The graph of f has a local maximum at x=a.

C) The graph of f has a local minimum at x=a.

D) The graph of f does not have a point of inflection at x=a.

Correct Answer: D

A point of inflection occurs where the second derivative changes sign. Here, f''(x) = (x-a)^2. While f''(a) = 0, the term (x-a)^2 is greater than or equal to zero for all x. This means f''(x) does not change sign at x=a. Therefore, the graph of f does not have a point of inflection at x=a.

The behavior of which derivative is used to directly determine the intervals of upward or downward concavity for a function f?

A) The first derivative, f'

B) The second derivative, f''

C) The third derivative, f'''

D) The original function, f

Correct Answer: B

The provided content explicitly states that the second derivative of a function provides information about the function and its graph, including intervals of upward or downward concavity. The sign of the second derivative indicates the direction of concavity.