AP Calculus BC · Unit 5: Analytical Applications of Differentiation
5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
In short
You learn that existence theorems allow us to draw conclusions about a function’s behavior on an interval without precisely locating that behavior. The Extreme Value Theorem guarantees that if a function is continuous over a closed interval [a, b], it has at least one minimum and one maximum value on that interval.
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What else do you need to know about Extreme Value Theorem, Global Versus Local Extrema, and Critical Points?
You must understand that a critical point occurs where the first derivative equals zero or fails to exist. You must also recognize that while all local extrema occur at critical points, not all critical points are local extrema. The underlying processes of finding critical points and extrema are the foundation for justifications. You should use calculators to graph a function and its derivatives to explore related features and confirm calculations.
Can you do these?
- I can apply the Extreme Value Theorem to guarantee a minimum and maximum value for a continuous function on a closed interval.
- I can find critical points of a function by determining where its first derivative is zero or undefined.
- I can explain that all local extrema occur at critical points, but not all critical points result in local extrema.
How is Extreme Value Theorem, Global Versus Local Extrema, and Critical Points tested on the exam?
On the AP Exam, sound reasoning must be accompanied by clear communication. Many students struggle with misinterpreting the characteristics of the graph of a derivative as though they are characteristics of the original function, or they use nonspecific language (e.g., 'it' rather than 'f'). You are expected to refer to f, f', and f'' by name. You should avoid relying on insufficient evidence or descriptions in their justifications, such as stating 'the graph of f is increasing because it’s going up.' Instead, you must provide model calculus-based justifications based on the analysis of a derivative. When answering questions about relative extrema, using the language in the question as a starting point (e.g., 'g has neither a relative maximum nor a relative minimum at x = 10, because...') is a recommended strategy.
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Extreme Value Theorem, Global Versus Local Extrema, and Critical Points: common questions
What do you need to know about Extreme Value Theorem, Global Versus Local Extrema, and Critical Points for AP Calculus BC?
You learn that existence theorems allow us to draw conclusions about a function’s behavior on an interval without precisely locating that behavior. The Extreme Value Theorem guarantees that if a function is continuous over a closed interval [a, b], it has at least one minimum and one maximum value on that interval.
How is Extreme Value Theorem, Global Versus Local Extrema, and Critical Points tested on the AP Calculus BC exam?
On the AP Exam, sound reasoning must be accompanied by clear communication. Many students struggle with misinterpreting the characteristics of the graph of a derivative as though they are characteristics of the original function, or they use nonspecific language (e.g., 'it' rather than 'f').
How do I know I have mastered Extreme Value Theorem, Global Versus Local Extrema, and Critical Points?
Check you can do each of these: apply the Extreme Value Theorem to guarantee a minimum and maximum value for a continuous function on a closed interval; find critical points of a function by determining where its first derivative is zero or undefined; explain that all local extrema occur at critical points, but not all critical points result in local extrema.
Where does Extreme Value Theorem, Global Versus Local Extrema, and Critical Points fit in AP Calculus BC?
Extreme Value Theorem, Global Versus Local Extrema, and Critical Points is topic 5.2 in Unit 5: Analytical Applications of Differentiation of AP Calculus BC.
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