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AP Calculus AB Practice Quiz: Extreme Value Theorem, Global Versus Local Extrema, and Critical Points

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

A function g is continuous on the closed interval [-1, 5]. What does the Extreme Value Theorem guarantee for g on this interval?

All Questions (10)

A function g is continuous on the closed interval [-1, 5]. What does the Extreme Value Theorem guarantee for g on this interval?

A) g has at least one absolute minimum and one absolute maximum value.

B) g has at least one critical point.

C) The derivative of g is zero at some point in (-1, 5).

D) g is differentiable on the open interval (-1, 5).

Correct Answer: A

According to the Extreme Value Theorem, if a function is continuous over a closed interval [a, b], it is guaranteed to have at least one minimum value and one maximum value on that interval. [cite: 2486]

For which of the following scenarios does the Extreme Value Theorem NOT guarantee a maximum and minimum value for the function f?

A) f(x) = x^2 on the interval [-2, 2]

B) f(x) = |x| on the interval [-1, 1]

C) A function f that is continuous on the open interval (0, 4).

D) A function f that is continuous on the closed interval [1, 100].

Correct Answer: C

The Extreme Value Theorem requires the function to be continuous on a closed interval, such as [a, b]. An open interval (a, b) does not satisfy the conditions of the theorem. [cite: 2486]

A value c in the domain of a function f is defined as a critical point of f if which of the following is true?

A) f(c) = 0

B) f(c) is a local maximum or a local minimum.

C) The function f is not continuous at x=c.

D) f'(c) = 0 or f'(c) is undefined.

Correct Answer: D

By definition, a critical point of a function is a point on the function where the first derivative is equal to zero or fails to exist. [cite: 2487]

Let f be a function. If f has a local (relative) minimum at x = c, which of the following statements must be true?

A) f'(c) must be equal to zero.

B) x = c is a critical point of f.

C) The Extreme Value Theorem applies to f at the point x=c.

D) The function f must be continuous on a closed interval containing c.

Correct Answer: B

All local (relative) extrema occur at critical points of a function. A critical point is where the derivative is zero or undefined, so stating it's a critical point is the most complete and accurate conclusion. [cite: 2488]

A function h has a critical point at x = 2. Which of the following conclusions can be drawn with certainty?

A) h must have a local extremum at x = 2.

B) The Extreme Value Theorem guarantees h has a global extremum at x = 2.

C) h'(2) = 0 or h'(2) is undefined.

D) h must be discontinuous at x = 2.

Correct Answer: C

This question tests the definition of a critical point and the relationship between critical points and extrema. The definition of a critical point is that the derivative is zero or undefined. It is not guaranteed to be a local extremum. [cite: 2487, 2488]

A student is asked to find the absolute maximum value of a continuous function f on the interval [0, 10]. Which theorem justifies the conclusion that such a value must exist?

A) The Intermediate Value Theorem

B) The Mean Value Theorem

C) The Extreme Value Theorem

D) The Squeeze Theorem

Correct Answer: C

The Extreme Value Theorem is the theorem that justifies conclusions about the existence of absolute extrema for a continuous function on a closed interval. [cite: 2485, 2486]

Let f be a function such that f'(5) does not exist. Based on this information, what can be concluded about the point x = 5?

A) x = 5 is a local minimum of f.

B) x = 5 is a local maximum of f.

C) x = 5 is a critical point of f.

D) The Extreme Value Theorem cannot be applied to any interval containing x=5.

Correct Answer: C

The definition of a critical point is a point where the first derivative equals zero or fails to exist. Since f'(5) fails to exist, x=5 is a critical point of f. It may or may not be an extremum. [cite: 2487]

A function f is continuous on [a, b]. By the Extreme Value Theorem, f has a global maximum on this interval. Which statement accurately describes the location of this maximum?

A) The maximum must occur at a point c where f'(c) = 0.

B) If the maximum occurs at a point c in the open interval (a, b), then c must be a critical point.

C) The maximum must occur at a critical point.

D) The maximum cannot occur at an endpoint a or b.

Correct Answer: B

A global maximum that occurs in the interior of an interval (not at an endpoint) is also a local maximum. All local extrema occur at critical points. Therefore, if the global maximum is at c in (a, b), c must be a critical point. The maximum could also occur at an endpoint. [cite: 2488]

Let f be a function defined on an interval I. Which of the following statements is NOT always true?

A) If f is continuous on a closed interval [a, b], it must attain a maximum value on [a, b].

B) If f has a local maximum at x=c, then x=c is a critical point.

C) If x=c is a critical point of f, then f has a local extremum at x=c.

D) A critical point occurs where the first derivative is zero or is undefined.

Correct Answer: C

While all local extrema occur at critical points, not all critical points are local extrema. For example, for f(x) = x^3, x=0 is a critical point (f'(0)=0), but it is not a local maximum or minimum. [cite: 2488]

The Extreme Value Theorem guarantees a minimum and maximum value for a function f under which of the following primary conditions?

A) The function f is differentiable on the interval (a, b).

B) The function f has at least one critical point in the interval [a, b].

C) The function f is continuous on the closed interval [a, b].

D) The function f is defined for all real numbers.

Correct Answer: C

The two conditions required for the Extreme Value Theorem to apply are that the function must be continuous and the interval must be closed. [cite: 2486]