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AP Calculus BC Practice Quiz: Connecting a Function, Its First Derivative, and Its Second Derivative

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 differentiable function. If f'(x) < 0 for all x in the interval (a, b), which of the following must be true about f on (a, b)?

All Questions (7)

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

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 has a local minimum.

Correct Answer: B

The first derivative test states that if the first derivative of a function is negative on an interval, the function is decreasing on that interval. The sign of f'(x) determines whether f(x) is increasing or decreasing.

Let f be a twice-differentiable function such that f'(5) = 0 and f''(5) > 0. What can be concluded about the function f at x = 5?

A) f has a local minimum at x = 5.

B) f has a local maximum at x = 5.

C) f has a point of inflection at x = 5.

D) f is decreasing at x = 5.

Correct Answer: A

This is an application of the Second Derivative Test for local extrema. Since the first derivative is zero at x = 5, it is a critical point. Since the second derivative is positive, the graph of the function is concave up at x = 5, which indicates a local minimum.

The function f is differentiable, and its derivative, f', is increasing on the interval (3, 8). Which of the following statements must be true about the graph of f on the interval (3, 8)?

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

The concavity of the graph of f is determined by the sign of its second derivative, f''. The behavior of f' (whether it is increasing or decreasing) is also determined by the sign of its derivative, which is f''. Since f' is increasing, its derivative, f'', must be positive. A positive second derivative implies that the graph of f is concave up.

The graph of f', the derivative of a function f, is shown. The graph of f' has a local minimum at x = 1 and a zero at x = 3. At which value of x does the graph of f have a point of inflection?

A) x = 1

B) x = 3

C) Both x = 1 and x = 3

D) There is not enough information to determine.

Correct Answer: A

A point of inflection on the graph of f occurs where the concavity changes, which means f'' changes sign. A change in the sign of f'' corresponds to a point where f' changes from increasing to decreasing or vice versa. This occurs at a local extremum (maximum or minimum) of f'. The graph of f' has a local minimum at x = 1, so f has a point of inflection at x = 1. The point x=3 is where f' is zero, which corresponds to a local extremum for f.

The derivative of a function f is given by f'(x) = (x-1)^2(x-4). At what value of x does the graph of f have a local minimum?

A) x = 1

B) x = 4

C) Both x = 1 and x = 4

D) f has no local minimum.

Correct Answer: B

To find local extrema, we find the critical points by setting f'(x) = 0, which gives x=1 and x=4. We then analyze the sign of f'(x) around these points using the First Derivative Test. For x < 1, f'(x) is negative. For 1 < x < 4, f'(x) is also negative. Since the sign does not change at x=1, it is not a local extremum. For x > 4, f'(x) is positive. At x=4, the derivative changes from negative to positive, indicating a local minimum.

A point of inflection on the graph of a twice-differentiable function f occurs at a point where which of the following is true?

A) f' is zero or undefined.

B) f' is positive.

C) f'' changes sign.

D) f'' is positive.

Correct Answer: C

By definition, a point of inflection is a point on a curve at which the concavity changes. The concavity of f is determined by the sign of the second derivative, f''. Therefore, a point of inflection occurs where f'' changes from positive to negative or from negative to positive.

The graph of f', the derivative of a function f, is shown. On which of the following intervals is the graph of f both increasing and concave down?

A) (-3, -1)

B) (-1, 2)

C) (2, 4)

D) (4, 5)

Correct Answer: B

For the graph of f to be increasing, its derivative f' must be positive (i.e., the graph of f' must be above the x-axis). This occurs on the interval (-3, 4). For the graph of f to be concave down, its second derivative f'' must be negative, which means f' must be decreasing. The graph of f' is decreasing on the interval (-1, 2). The interval where both conditions are met is the intersection of (-3, 4) and (-1, 2), which is (-1, 2).