Absolute vs. Relative Maxima and Minima

In summary, the conversation discusses the relationship between absolute and relative extrema and how to determine if a point is both an absolute and relative extrema. It also mentions the method for finding absolute and relative extrema on a closed interval.
  • #1
Neen87
8
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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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  • #2
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.
 
  • #3
Thanks for your response!
 

FAQ: Absolute vs. Relative Maxima and Minima

1. What is the difference between absolute and relative maxima and minima?

Absolute maxima and minima are the highest and lowest values of a function over its entire domain, while relative maxima and minima are the highest and lowest values within a specific interval of the function.

2. How do you identify absolute and relative maxima and minima on a graph?

Absolute maxima and minima are identified by finding the highest and lowest points on the graph, respectively. Relative maxima and minima are identified by finding where the slope of the graph changes from positive to negative (for maxima) or negative to positive (for minima).

3. Can a function have multiple absolute or relative maxima or minima?

Yes, a function can have multiple absolute or relative maxima or minima. This can occur when there are multiple peaks or valleys in the graph of the function.

4. How are absolute and relative maxima and minima used in real-world applications?

Absolute and relative maxima and minima are used in optimization problems, such as finding the maximum profit or minimum cost in business or engineering scenarios. They can also be used in data analysis to identify the highest and lowest values in a dataset.

5. Is there a relationship between absolute and relative maxima and minima?

Yes, there is a relationship between absolute and relative maxima and minima. Relative maxima and minima are always included within the range of absolute maxima and minima. In other words, the absolute maxima and minima are the highest or lowest values that the function can attain, while the relative maxima and minima are the highest or lowest values within a smaller interval of the function.

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