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AP Calculus AB 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

If the first derivative of a function, f'(x), is negative on the interval (a, b), which of the following must be true about the function f(x) on that same interval?

All Questions (7)

If the first derivative of a function, f'(x), is negative on the interval (a, b), which of the following must be true about the function f(x) on that same interval?

A) f(x) is increasing.

B) f(x) is decreasing.

C) f(x) is concave up.

D) f(x) is concave down.

Correct Answer: B

The sign of the first derivative, f'(x), determines whether the function f(x) is increasing or decreasing. A negative first derivative (f'(x) < 0) on an interval indicates that the function f(x) is decreasing on that interval.

The second derivative of a function, f''(x), is positive for all x in the interval (c, d). What does this imply about the graph of the first derivative, f'(x), on this interval?

A) The graph of f'(x) is decreasing.

B) The graph of f'(x) is below the x-axis.

C) The graph of f'(x) is increasing.

D) The graph of f'(x) is above the x-axis.

Correct Answer: C

The second derivative, f''(x), is the derivative of the first derivative, f'(x). Just as a positive first derivative indicates that the original function is increasing, a positive second derivative indicates that the first derivative function, f'(x), is increasing.

A function f(x) is continuous and differentiable. If f'(2) = 0 and f''(2) < 0, what can be concluded about the function f(x) at x = 2?

A) f(x) has a local minimum at x = 2.

B) f(x) has a point of inflection at x = 2.

C) f(x) has a local maximum at x = 2.

D) The behavior of f(x) cannot be determined at x = 2.

Correct Answer: C

This is an application of the Second Derivative Test. When the first derivative is zero at a point (indicating a horizontal tangent), the sign of the second derivative determines the nature of the extremum. Since f'(2) = 0 and f''(2) < 0 (indicating the function is concave down), f(x) has a local maximum at x = 2.

For a function f(x), it is known that f'(x) < 0 and f''(x) > 0 on an interval. Which of the following describes the behavior of the graph of f(x) on this interval?

A) Increasing and concave up

B) Decreasing and concave up

C) Increasing and concave down

D) Decreasing and concave down

Correct Answer: B

The sign of the first derivative determines if the function is increasing or decreasing. Since f'(x) < 0, the function f(x) is decreasing. The sign of the second derivative determines the concavity. Since f''(x) > 0, the function f(x) is concave up. Therefore, the function is decreasing and concave up.

The graph of f'(x), the derivative of a function f(x), changes from decreasing to increasing at x = c. Which of the following must be true about the graph of f(x) at x = c?

A) f(x) has a local maximum.

B) f(x) has a local minimum.

C) f(x) has a point of inflection.

D) f(x) has a zero.

Correct Answer: C

A point of inflection on the graph of f(x) occurs where the concavity changes. The concavity of f(x) is determined by the sign of f''(x). The sign of f''(x) determines whether f'(x) is increasing or decreasing. If f'(x) changes from decreasing to increasing, it means f''(x) changes from negative to positive. This change in the sign of the second derivative corresponds to a point of inflection on the graph of f(x).

The graph of f'(x), the derivative of f(x), has a local maximum at x = a. What feature must the graph of the second derivative, f''(x), have at x = a?

A) A local maximum

B) A point of inflection

C) A zero

D) A local minimum

Correct Answer: C

If f'(x) has a local maximum at x = a, its derivative must be zero at that point (assuming the derivative exists). The derivative of f'(x) is f''(x). Therefore, f''(a) must be equal to 0. This corresponds to a zero (an x-intercept) on the graph of f''(x).

A function f(x) has a point of inflection at x = 5. Which of the following statements about its derivatives, f'(x) and f''(x), could be true?

A) f'(5) = 0 and f''(5) > 0

B) f'(x) is constant around x = 5.

C) f''(x) has a local extremum at x = 5.

D) f'(x) has a local extremum at x = 5.

Correct Answer: D

A point of inflection on f(x) occurs where the concavity changes, which means f''(x) changes sign. For f''(x) to change sign at x = 5, it must be that f''(5) = 0 or is undefined. If f''(5) = 0, this means that the derivative of f'(x) is zero at x = 5, which is the condition for f'(x) to have a critical point. This critical point is often a local extremum (maximum or minimum) on the graph of f'(x).