AP Calculus BC Flashcards: Using the Candidates Test to Determine Absolute (Global) Extrema
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Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
On a closed interval, where can the absolute (global) extrema of a function occur?
The absolute (global) extrema of a function on a closed interval can only occur at the function's critical points or at the interval's endpoints.
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On a closed interval, where can the absolute (global) extrema of a function occur?
The absolute (global) extrema of a function on a closed interval can only occur at the function's critical points or at the interval's endpoints.
What is the first step in applying the Candidates Test to find absolute extrema on a closed interval?
The first step is to use the function's derivative to find all critical points that lie within the specified closed interval.
What is the primary purpose of the Candidates Test?
The Candidates Test is a method for finding the absolute (global) maximum and minimum values of a continuous function on a closed interval.
A function is continuous on [0, 10] and has critical points at x=2 and x=12. What are the candidates for its absolute maximum?
The candidates are the endpoints (x=0 and x=10) and the critical point that is within the interval (x=2).
Why must endpoints be checked when finding absolute extrema on a closed interval?
Endpoints must be checked because the highest or lowest point on a specific interval can occur at the boundary, even if it is not a critical point.
To find the absolute minimum of a function on [a, b], what values must you ultimately compare?
You must compare the function's output values (y-values) at the endpoints (f(a) and f(b)) and at all critical points within the interval.
What condition is necessary for the Candidates Test to guarantee the existence of absolute extrema?
The function must be continuous on a closed interval for the test to guarantee that an absolute extremum exists at a critical point or an endpoint.
What are the "candidates" in the Candidates Test?
The candidates are the specific x-values where an absolute extremum could occur: the endpoints of the closed interval and all critical points within that interval.
How is the behavior of a function's derivatives used to justify conclusions about extrema?
A function's derivatives are used to locate critical points (where the derivative is zero or undefined), which are potential locations for extrema.
After finding the derivative of a function, what is the next step in identifying candidates for absolute extrema?
After finding the derivative, you must determine where the derivative is equal to zero or is undefined to locate the critical points.