Extreme Value Theorem true for constants?

In summary, the conversation discusses the extreme value theorem and its application to constant functions. The person expressing their frustration with the theorem argues that for a constant function, every point is a minimum and maximum, which they find unimpressive. The other person explains that the theorem simply states that a function must have a minimum and maximum on a closed interval, but they do not have to be different or at every point. They also define the maximum and minimum of a function as the highest and lowest values it reaches. Despite understanding the logic, the person is still not pleased with the theorem.
  • #1
member 508213
My textbook says the extreme value theorem is true for constants but I don't buy it. I mean I suppose that every value over a closed interval for a constant would be a maximum and a minimum technically but it seems like BS to me. Can anyone explain why this BS is true?
 
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  • #2
Constant functions are continuous, I don't really understand what your issue is. Can you elaborate about why this is bs?
 
  • #3
No, my problem is the extreme value theorem says that for a continuous function on a closed interval there will be a minimum and maximum value. My complaint with that is for a constant every point is a minimum and a maximum and I just think that is pretty lame so I'm mad.
 
  • #4
Austin said:
My complaint with that is for a constant every point is a minimum and a maximum and I just think that is pretty lame so I'm mad.
Well, okay then.
 
  • #5
Well thanks for your answer, my problem isn't with my understanding of it I just don't really like it. However, your point about the theorem saying that the function must simply have a min and max makes sense, I suppose that the min and max don't have to be different and not be at every point; I still don't like it though.
 
  • #6
As a follow up question, how would you define the maximum and minimum of a function? Do you know the definition or do you just find it is not pleasing to how one thinks about maximum and minimum?
 
  • #7
Max= highest value the function reaches
Min= lowest value a function reaches

So for a constant yes they would have both technically. Like I said I just think it's poop.
 
  • #8
But I do see what you're saying and I appreciate your logic. Thank you.
 
  • #9
Hmm, Alright.
 
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  • #10
Did you get my message?
 

1. Is the Extreme Value Theorem true for all constants?

Yes, the Extreme Value Theorem is true for all constants. This theorem states that a continuous function on a closed interval will have an absolute maximum and minimum value on that interval, regardless of the value of the constant.

2. What is the significance of the Extreme Value Theorem?

The Extreme Value Theorem is important because it guarantees the existence of maximum and minimum values for continuous functions on closed intervals. This allows us to make accurate predictions and calculations in real-life situations, as well as prove important mathematical theorems.

3. Can the Extreme Value Theorem be applied to functions with multiple variables?

Yes, the Extreme Value Theorem can be extended to functions with multiple variables. In this case, the theorem states that a continuous function on a closed and bounded region in multiple dimensions will have both an absolute maximum and minimum value.

4. How is the Extreme Value Theorem related to the Mean Value Theorem?

The Mean Value Theorem is a special case of the Extreme Value Theorem. It states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists a point within the interval where the derivative of the function is equal to the average rate of change of the function on that interval.

5. Can the Extreme Value Theorem be used to find the exact values of the maximum and minimum points?

No, the Extreme Value Theorem only guarantees the existence of maximum and minimum values, but it does not provide a method for finding their exact values. Additional techniques, such as differentiation or optimization, are needed to determine the exact values of these points.

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