Relative Extrema vs Absolute Extrema

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SUMMARY

The discussion clarifies the distinction between relative extrema and absolute extrema in mathematical functions. A relative maximum or minimum is defined as the greatest or smallest value within a specific neighborhood, while an absolute maximum or minimum represents the greatest or smallest value across the entire domain. The conversation highlights that to determine absolute extrema, one must evaluate critical points, poles, and the boundaries of the function's domain. The example provided illustrates a function with three relative minima and one absolute minimum, with no absolute maximum due to the function approaching infinity at the boundaries.

PREREQUISITES
  • Understanding of critical points in calculus
  • Familiarity with function behavior and limits
  • Knowledge of mathematical terminology related to extrema
  • Ability to analyze graphical representations of functions
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  • Study the process of finding critical points in calculus
  • Learn about limits and their role in determining function behavior
  • Explore graphical methods for identifying relative and absolute extrema
  • Investigate the implications of poles in function analysis
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Students studying calculus, educators teaching mathematical concepts, and anyone interested in understanding the behavior of functions in relation to extrema.

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In basic terms, what is the main difference between relative extrema and absolute extrema? I know that absolute extrema is more involved but why is this the case?
 
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A relative maximum (minimum) is the greatest (smallest) in its neighborhood but an absolute maximum (minimum) is the greatest (smallest) anywhere (in the domain). For example:

View attachment 6470

In the picture we see that the function has $3$ relative minima and an absolute minimum, which is smallest of all relative minima. It also has $2$ relative maxima, but it has no absolute maximum, since at the boundaries the function goes to $+\infty$.
In general, to find an absolute extrema, besides the critical points we have to check also the poles of the function and the boundaries of its domain.
 

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mathmari said:
A relative maximum (minimum) is the greatest (smallest) in its neighborhood but an absolute maximum (minimum) is the greatest (smallest) anywhere (in the domain). For example:



In the picture we see that the function has $3$ relative minima and an absolute minimum, which is smallest of all relative minima. It also has $2$ relative maxima, but it has no absolute maximum, since at the boundaries the function goes to $+\infty$.
In general, to find an absolute extrema, besides the critical points we have to check also the poles of the function and the boundaries of its domain.

Very cool. Nice picture.
 

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