



Why are boundaries important in finding extrema of multivariable functions?
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what is the extreme value theorem Informational
- Does it play a role in the study of stability?
- Is there a connection to fixed points of set-valued maps?
- What is the connection to the least upper bound and greatest lower bound?
- Discuss the impact of different discontinuities. Can a function have local extrema but no absolute extrema on a closed interval?
- Illustrate a physics application. Can you give an example of using the Extreme Value Theorem in engineering?
- Are there analogues in the realm of quantum mechanics and gravity?
- How can it be used to show that an equation has a solution?
- How do Lagrange multipliers relate to the Extreme Value Theorem? How does the Extreme Value Theorem connect to the concept of level sets?
- Discuss its relevance to the calculus of variations. How can the Extreme Value Theorem be used to analyze the stability of equilibrium points in dynamical systems?
- Give examples of real-world uses of the Extreme Value Theorem. Can you explain the proof of the Extreme Value Theorem?
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