Why is there no absolute min / max for the given equation?

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In summary, the given equation f(x) = 3x^2 / x - 3 on [2,8] has critical points at [3, 5/2, 0] and has a derivative of f ' (x) = 3x^2 (x/-5) / (x-3)^2. There is no absolute min/max in this case because the interval [2,8] is out of the bounds of the critical points. The function is not continuous on the interval [2,8] and therefore may have only an absolute minimum, absolute maximum, or neither. The critical point at 3 is not considered because it is not in the domain of the function.
  • #1
jwxie
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Given equation
f(x) = 3x^2 / x - 3 on [2,8]

I found the critical points were [3, 5/2, 0]
f ' (x) = 3x^2 (x/-5) / (x-3)^2

I want to know why there is no absolute min / max in this case?

I think the reason is because this [a,b] is out of the bound of the critical point, thus we cannot compare [a,b] against the critical points?

Also, is 0 the critical point?
f ' (x) = 3x^2 (x/-5) / (x-3)^2
3x^2 = 0
so i got 0
i always got confused with 1/2 =/ 0
 
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  • #2
When you write an equation like this, use parentheses. I.e. f(x)=3x2/(x-3).

Is this function continuous on the interval [2,8]? If it is, then it must have an absolute maximum and minimum in the interval [2,8], but if not, it may have only an absolute minimum, absolute maximum, or neither.
 
  • #3
Thanks for the reminder.
Oh right... 3 is undefined.

Oh I did not notice 3 was also my C.P (dumb)
Thanks!
 
  • #4
Right! Although, 3 isn't a critical point because it isn't in the domain of the function. Just take the limits as x approaches 3 from the left and right hand sides to show that the function is unbounded in the interval [2,8].
 

FAQ: Why is there no absolute min / max for the given equation?

Why is there no maximum or minimum in science?

The concept of maximum and minimum values is often used in mathematics to describe the boundaries of a function or equation. However, in science, we are dealing with real-world phenomena that are constantly changing and evolving. As a result, there is no specific maximum or minimum value that can be applied universally.

Can't scientists determine a maximum or minimum through experimentation?

In some cases, scientists can determine a maximum or minimum value through experimentation. However, this value is often limited to a specific set of conditions and may not be applicable to other scenarios. Additionally, due to the complexity of many scientific phenomena, it may not be possible to accurately determine a definitive maximum or minimum value.

How do scientists work with data if there is no maximum or minimum?

Scientists work with data by analyzing trends and patterns rather than focusing on specific maximum or minimum values. By studying the relationships between variables and identifying trends, scientists can make informed conclusions and predictions about the behavior of a system or phenomenon.

Is it possible that there is a maximum or minimum value that we just haven't discovered yet?

It is always possible that there is a maximum or minimum value that has not been discovered or fully understood yet. As science and technology continue to advance, our understanding of the world around us also evolves. Therefore, what may have been considered the maximum or minimum in the past may not hold true in the future.

How does the lack of a maximum or minimum value affect scientific theories and models?

The lack of a maximum or minimum value does not necessarily affect scientific theories and models. In fact, many scientific theories and models are based on the idea of continuous change rather than fixed values. As long as the predictions and explanations provided by a theory or model are supported by evidence and can be tested and refined, the lack of a maximum or minimum value does not invalidate its validity.

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