Absolute Maxima and absolute minima

 Absolute maxima and absolute minima are special cases of maxima and minima in mathematics. They represent the highest and lowest values of a function, respectively, over its entire domain. These points are referred to as "absolute" because they are not limited to specific intervals but encompass the entire range of the function.

  1. Absolute Maximum (Global Maximum): The absolute maximum of a function is the highest value that the function attains over its entire domain. Mathematically, if f(x) is a function defined over a domain D, then a point (c, f(c)) is considered an absolute maximum if for all x in D, f(c) ≥ f(x).


  3. Absolute Minimum (Global Minimum): The absolute minimum of a function is the lowest value that the function attains over its entire domain. Mathematically, if f(x) is a function defined over a domain D, then a point (c, f(c)) is considered an absolute minimum if for all x in D, f(c) ≤ f(x).

To find the absolute maxima and minima of a function, you generally follow these steps:

  1. Identify the domain of the function.
  2. Find critical points by setting the derivative equal to zero and finding points where the derivative is undefined within the domain.
  3. Evaluate the function at these critical points and at the endpoints of the domain.
  4. The highest value corresponds to the absolute maximum, and the lowest value corresponds to the absolute minimum.

It's important to note that not all functions have an absolute maximum or an absolute minimum. If the function is unbounded (i.e., it increases or decreases without bound), it may not have an absolute maximum or minimum. However, if the function is continuous and defined over a closed and bounded interval (a compact set), the absolute maximum and minimum are guaranteed to exist due to the extreme value theorem.

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