AP Calculus BC Flashcards: Using the First Derivative Test to Determine Relative (Local) 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.
Why is a sign change in the first derivative necessary to identify a relative extremum?
A sign change in the derivative signifies that the function's behavior has switched from increasing to decreasing (a maximum) or from decreasing to increasing (a minimum).
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Why is a sign change in the first derivative necessary to identify a relative extremum?
A sign change in the derivative signifies that the function's behavior has switched from increasing to decreasing (a maximum) or from decreasing to increasing (a minimum).
A function's derivative is f'(x) = x(x-4). Justify the conclusion about the function's behavior at x=0.
The function f(x) has a relative maximum at x=0 because its derivative, f'(x), changes from positive to negative at that point.
If the derivative f'(x) is positive before a critical point x=c and also positive after x=c, what can you conclude about the function at that point?
You can conclude that there is no relative extremum at x=c because the derivative does not change signs.
What does the first derivative of a function, f'(x), reveal about the behavior of the original function, f(x)?
The behavior of the first derivative determines the behavior of the function; for example, the sign of f'(x) indicates whether f(x) is increasing or decreasing.
What is the primary use of the First Derivative Test?
The First Derivative Test uses the sign changes of the first derivative to determine the location of relative (local) extrema of a function.
What is the relationship between the behavior of a function and the behavior of its derivatives?
The behavior of a function, such as where it is increasing, decreasing, or has extrema, can be determined by analyzing the behavior of its derivatives.
How do you justify that a function f(x) has a relative maximum at x=c using the First Derivative Test?
A relative maximum at x=c is justified by stating that the derivative, f'(x), changes from positive to negative at that point.
What is a relative (or local) extremum?
A relative (or local) extremum is a point on a function that is either a maximum or a minimum value compared to the points immediately surrounding it.
What must be identified first before the First Derivative Test can be applied to find relative extrema?
You must first find the function's critical points, which are the locations where the derivative is either zero or undefined.
How do you justify that a function f(x) has a relative minimum at x=c using the First Derivative Test?
A relative minimum at x=c is justified by stating that the derivative, f'(x), changes from negative to positive at that point.