


Explain the connection to convex functions. What are some examples of functions where the absolute extrema occur at points where the derivative does not exist, even on a closed interval?
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what is the extreme value theorem Informational
- Explain the topological connection. What are some applications of the Extreme Value Theorem in the study of partial differential equations?
- Discuss its relevance to measure and probability. How does the Extreme Value Theorem connect to the idea of manifolds with boundary in differential topology?
- Illustrate extrema at non-smooth points.
- Explain the relationship. Can you provide a visual analogy to explain the Extreme Value Theorem to someone without a strong math background?
- Illustrate extrema on disconnected sets.
- Explain the relationship. What are some variations of the Extreme Value Theorem for semicontinuous functions?
- Illustrate extrema on infinite closed sets.
- Explore the reverse implication of the Extreme Value Theorem. Can you have absolute extrema without the interval being closed?
- Provide such examples. How does the Extreme Value Theorem relate to the concept of a Banach space or Hilbert space in functional analysis?
- Discuss applications in financial modeling. How does the Extreme Value Theorem connect to the idea of a quantum manifold or quantum space in modern theoretical physics?
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