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Intro to Extrema quiz #1

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  • What is the difference between global (absolute) extrema and local (relative) extrema in a function?
    Global extrema are the highest and lowest points over the entire function, while local extrema are the highest and lowest points within a specific region or neighborhood of the function.
  • How is a global maximum of a function defined mathematically?
    A global maximum occurs at x = c if f(c) ≥ f(x) for all x in the domain of the function.
  • How is a local maximum of a function defined mathematically?
    A local maximum occurs at x = c if f(c) ≥ f(x) for all x near c (in some open interval around c).
  • Can a point be both a global and a local extremum? Explain.
    Yes, a point can be both a global and a local extremum if it is the highest or lowest point in the entire function and also in its immediate region.
  • According to the course convention, can endpoints of a function be considered local extrema?
    No, endpoints can be global extrema but not local extrema according to the course convention.
  • What is another term for global extrema and local extrema?
    Global extrema are also called absolute extrema, and local extrema are also called relative extrema.
  • Why is it important to distinguish between global and local extrema when analyzing a function?
    Distinguishing between global and local extrema helps in understanding the overall behavior of a function and identifying the highest and lowest values both globally and within specific regions.
  • If a function has a minimum at an endpoint, is it always considered a global minimum?
    It is considered a global minimum if it is the lowest value of the entire function, but it is not considered a local minimum if it occurs at an endpoint.
  • What is the main difference between global (absolute) extrema and local (relative) extrema in a function?
    Global extrema are the highest and lowest points over the entire function, while local extrema are the highest and lowest points within a specific region or neighborhood of the function.
  • According to the course convention, can endpoints of a function be considered local extrema?
    No, endpoints can be global extrema but not local extrema according to the course convention.