AP Calculus AB Flashcards: Using the Second Derivative Test to Determine Extrema
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.
What two properties must a function have for its single relative extremum on an interval to also be an absolute extremum?
The function must be continuous on the interval and it must have only one critical point on that interval.
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What two properties must a function have for its single relative extremum on an interval to also be an absolute extremum?
The function must be continuous on the interval and it must have only one critical point on that interval.
What is the role of the second derivative in determining extrema?
The second derivative of a function may be used to determine whether a critical point is the location of a relative (local) maximum or minimum.
What is an absolute (global) extremum?
It is the overall highest or lowest point of a function on an interval, which can sometimes be found by identifying a single critical point that is a relative extremum.
On an interval, a continuous function has only one critical point, which is a relative maximum. What can you conclude about this point for the entire interval?
This single critical point also corresponds to the absolute (global) maximum of the function on that interval.
What is the fundamental relationship between a function's behavior and its derivatives?
Conclusions about the behavior of a function (like the location of extrema) can be justified by analyzing the behavior of its derivatives.
If you use the second derivative to classify a critical point as a local maximum, what general calculus principle are you demonstrating?
You are demonstrating the principle that the behavior of a function's derivatives can be used to justify conclusions about the behavior of the original function.
A continuous function has a single critical point on its entire domain, which is found to be a relative minimum. What else can you conclude about this point?
Because it is the only critical point for the continuous function, this relative minimum also corresponds to the absolute (global) minimum of the function.
Under what specific condition does a relative extremum also become an absolute extremum on an interval?
When a continuous function has only one critical point on an interval, and that point is a relative extremum, it is also the absolute extremum on that interval.
What is the primary purpose of using the Second Derivative Test?
Its primary purpose is to classify a critical point of a function as either the location of a relative maximum or a relative minimum.
What is the connection between a critical point and a relative extremum?
A critical point is a potential location for a relative extremum, and the second derivative can be used to determine if it is a maximum or minimum.