AP Calculus BC Flashcards: Using the Second Derivative Test to Determine Extrema
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Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
How do you justify the existence of a relative minimum at a critical point using derivatives?
You can justify a relative minimum by showing the first derivative is zero and the second derivative is positive at that point.
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How do you justify the existence of a relative minimum at a critical point using derivatives?
You can justify a relative minimum by showing the first derivative is zero and the second derivative is positive at that point.
A continuous function has only one critical point on an interval, and the Second Derivative Test confirms it's a relative minimum. What further conclusion can be made?
This single relative minimum must also be the absolute (global) minimum of the function on that interval.
What is the significance of a critical point in the context of the Second Derivative Test?
A critical point is a potential location for a relative extremum, which the second derivative can then help to classify as a maximum or minimum.
Define relative (local) extremum.
A relative (local) extremum is a point that is a maximum or minimum value of a function in its immediate vicinity or a small open interval.
A function has a single critical point on its domain, which is identified as a relative maximum. What is the global significance of this point?
This point also corresponds to the absolute (global) maximum of the function on its entire domain.
Using the Second Derivative Test, what can you conclude if the second derivative at a critical point is negative?
If the second derivative is negative at a critical point, the function has a relative (local) maximum at that point.
What is the role of the second derivative in finding 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.
Using the Second Derivative Test, what can you conclude if the second derivative at a critical point is positive?
If the second derivative is positive at a critical point, the function has a relative (local) minimum at that point.
What is the fundamental principle for analyzing a function's behavior using calculus?
Conclusions about the behavior of a function can be justified by analyzing the behavior of its derivatives.
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.