



Why are closed intervals important for the Extreme Value Theorem?
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what is the extreme value theorem Informational
- Explain the topological connection. What are some examples of functions where the absolute extrema are difficult to find even though the interval is closed and the function is continuous?
- Discuss its connection to functional analysis. What are some advanced topics in optimization theory that build upon the foundations provided by the Extreme Value Theorem?
- Differentiate between them. How can the Extreme Value Theorem be used to prove other theorems in calculus?
- Is it possible to have a global maximum on an open interval?
- What does it tell us about sequences that approach the extrema?
- If a function has absolute extrema on a closed interval, must it be continuous?
- Explain the connection. What are some examples of functions where the absolute extrema are not unique, but the values at which they occur form a set with non-zero measure?
- Provide a scenario where extrema are found at both an interior point and a boundary.
- Is there a connection between finding extrema and fixed points?
- Explain extensions to vector-valued functions. What are some practical strategies for finding absolute extrema when the Extreme Value Theorem applies?
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