Absolute vs. Relative Maxima and Minima

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SUMMARY

The discussion clarifies the relationship between absolute and relative maxima and minima in calculus. It establishes that while every absolute extremum is also a relative extremum, the reverse is not necessarily true. To identify absolute extrema, one must evaluate critical points where the derivative f'(x) equals zero or is undefined, along with the endpoints of the interval. This method allows for determining the overall maximum and minimum values effectively.

PREREQUISITES
  • Understanding of calculus concepts, specifically extrema.
  • Knowledge of critical points and their significance in function analysis.
  • Familiarity with the derivative and its role in identifying maxima and minima.
  • Ability to evaluate functions over closed intervals.
NEXT STEPS
  • Study the process of finding critical points in calculus.
  • Learn about the Mean Value Theorem and its implications for extrema.
  • Explore graphical methods for visualizing absolute and relative extrema.
  • Review examples of functions with both absolute and relative maxima/minima.
USEFUL FOR

Students studying calculus, educators teaching mathematical concepts, and anyone seeking to deepen their understanding of function behavior in relation to extrema.

Neen87
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Homework Statement



I do not have a specific homework problem, but could someone please clarify this for me?

QUESTION: When you have an absolute maxima (or minima), how can you tell if it is ALSO a relative maxima (or minima)?

I understand how to find absolute extrema on a closed interval, and how to find critical values of a function by setting f'(x) = 0 and when f'(x) is undefined.


Thank you!
Tina
 
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Neen87 said:

Homework Statement



I do not have a specific homework problem, but could someone please clarify this for me?

QUESTION: When you have an absolute maxima (or minima), how can you tell if it is ALSO a relative maxima (or minima)?

I understand how to find absolute extrema on a closed interval, and how to find critical values of a function by setting f'(x) = 0 and when f'(x) is undefined.


Thank you!
Tina
The question is usually asked the other way around. IOW, If you have a relative maximum (or minimum), how can you tell if it is ALSO an absolute maximum (minimum)?

It's generally easier to find relative maxima or minima by finding the values for which the derivative is zero or undefined and checking endpoints of the domain. From these points, it's just a matter of finding the overall largest and smallest to get the absolute maximum and minimum.
 
Thanks for your response!
 

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