AP Calculus BC Flashcards: Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
A function f(x) is continuous on the interval [-2, 10]. What conclusion can you make about its extrema on this interval?
By applying the Extreme Value Theorem, we can conclude that f(x) is guaranteed to have at least one absolute minimum and one absolute maximum value on [-2, 10].
Card 1 of 10
All Flashcards (10)
A function f(x) is continuous on the interval [-2, 10]. What conclusion can you make about its extrema on this interval?
By applying the Extreme Value Theorem, we can conclude that f(x) is guaranteed to have at least one absolute minimum and one absolute maximum value on [-2, 10].
What does the Extreme Value Theorem (EVT) guarantee?
The EVT guarantees that if a function is continuous over a closed interval [a, b], it must have at least one absolute minimum value and one absolute maximum value on that interval.
True or False: If a function has a local maximum at x=c, then x=c must be a critical point.
True. All local extrema, both maximums and minimums, occur at critical points.
Describe the relationship between local (relative) extrema and critical points.
All local extrema must occur at critical points of a function. However, not all critical points are necessarily local extrema.
If a function's derivative is zero at a point, is that point guaranteed to be a local extremum?
No, it is not guaranteed. While the point is a critical point, not all critical points are local extrema.
What theorem would you cite to justify that a continuous function must achieve a maximum value on a closed interval?
You would apply the Extreme Value Theorem to justify this conclusion.
Define a critical point of a function.
A critical point is a point on a function where the first derivative either equals zero or fails to exist.
What are the two ways the first derivative can indicate a critical point?
A critical point occurs where the first derivative is equal to zero or where the first derivative fails to exist.
What are the two required conditions for the Extreme Value Theorem to apply?
For the Extreme Value Theorem to apply, the function must be continuous and defined on a closed interval [a, b].
Where should you look for the absolute maximum and minimum values of a continuous function on a closed interval?
You should evaluate the function at all critical points within the interval and at the endpoints of the interval.