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AP Calculus BC Practice Quiz: Using the First Derivative Test to Determine Relative (Local) Extrema

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

The first derivative of a function, f'(x), is primarily used to determine which of the following properties of the function f(x)?

All Questions (7)

The first derivative of a function, f'(x), is primarily used to determine which of the following properties of the function f(x)?

A) The function's points of inflection.

B) The location of the function's relative (local) extrema.

C) The function's concavity.

D) The function's absolute maximum value on a closed interval.

Correct Answer: B

Based on the provided content, the first derivative of a function is used to determine the location of its relative (local) extrema by analyzing where the derivative changes sign.

Let f be a function with a first derivative given by f'(x) = x(x - 2). At what value of x does f have a relative minimum?

A) x = -2

B) x = 0

C) x = 1

D) x = 2

Correct Answer: D

The critical points of f are where f'(x) = 0, which are x=0 and x=2. For x < 0, f'(x) is positive. For 0 < x < 2, f'(x) is negative. For x > 2, f'(x) is positive. A relative minimum occurs where f'(x) changes from negative to positive. This change occurs at x = 2.

The function g has a critical point at x = 3. If the derivative g'(x) is positive for x < 3 and negative for x > 3, which of the following statements provides a correct justification for the behavior of g at x = 3?

A) g has a relative minimum at x = 3 because g'(x) changes from positive to negative.

B) g has a relative maximum at x = 3 because g'(x) changes from positive to negative.

C) g has a relative minimum at x = 3 because g'(x) changes from negative to positive.

D) g has a relative maximum at x = 3 because g'(x) changes from negative to positive.

Correct Answer: B

According to the First Derivative Test, a function has a relative maximum at a point where its first derivative changes sign from positive to negative. This indicates the function changes from increasing to decreasing.

Let f be a function such that its first derivative is f'(x) = (x + 1)^2(x - 4). Which of the following statements is true about the function f?

A) f has a relative maximum at x = -1 and a relative minimum at x = 4.

B) f has a relative minimum at x = 4 and no relative extremum at x = -1.

C) f has a relative maximum at x = 4 and a relative minimum at x = -1.

D) f has a relative minimum at x = -1 and no relative extremum at x = 4.

Correct Answer: B

The critical points are x = -1 and x = 4. For x < -1, f'(x) is negative. For -1 < x < 4, f'(x) is also negative. Since f'(x) does not change sign at x = -1, there is no relative extremum there. For x > 4, f'(x) is positive. Since f'(x) changes from negative to positive at x = 4, f has a relative minimum at x = 4.

The derivative of a function h is given by h'(x) = (x^2 - 9)e^x. On what intervals is the function h increasing?

A) (-3, 3)

B) (-∞, -3) and (3, ∞)

C) (-∞, 9)

D) (0, ∞)

Correct Answer: B

A function is increasing where its first derivative is positive. Since e^x is always positive, the sign of h'(x) is determined by the sign of (x^2 - 9). The expression x^2 - 9 is positive when x < -3 or x > 3. Therefore, h is increasing on the intervals (-∞, -3) and (3, ∞).

Let f be a function that is continuous on the interval [-2, 5] and differentiable on (-2, 5). If f'(-1) = 0 and f'(x) > 0 for all other x in (-2, 5), which of the following provides a valid conclusion?

A) f has a relative maximum at x = -1.

B) f has a relative minimum at x = -1.

C) f has neither a relative maximum nor a relative minimum at x = -1.

D) The First Derivative Test cannot be used to draw a conclusion at x = -1.

Correct Answer: C

For a relative extremum to exist at a critical point x = c, the first derivative f'(x) must change sign at x = c. In this case, f'(x) is positive on both sides of x = -1. Since the sign of f'(x) does not change, f does not have a relative extremum at x = -1.

The first derivative of the function g is given by g'(x) = (x-1)/(x+3). For what value(s) of x does g have a relative maximum?

A) x = 1 only

B) x = -3 only

C) x = 1 and x = -3

D) g has no relative maximum.

Correct Answer: D

Critical points occur where g'(x) = 0 or is undefined. g'(x) = 0 at x=1. g'(x) is undefined at x=-3. For x < -3, g'(x) is positive. For -3 < x < 1, g'(x) is negative. For x > 1, g'(x) is positive. A relative maximum occurs where g' changes from positive to negative, which happens at x = -3. However, since g' is undefined at x=-3, g is not differentiable there and may have a cusp or vertical asymptote, but the question asks for a relative maximum based on the derivative test. The change from negative to positive at x=1 indicates a relative minimum. Therefore, there is no relative maximum.