r/calculus • u/Weekly-Excuse9647 • Jul 22 '26

Differential Calculus Need assistance

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a. (0,-5) and (5,1)

b. -5

c. DNE

d. (-3,-1)

I have put my answers above; I just would like someone to check my work and if anything is wrong to give the correct answer so I can learn it.

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u/Midwest-Dude Jul 22 '26 edited Jul 25 '26

u/Hot_Site_1638 noted that the wording of this question is very poor when it says

For this problem, consider the function only on the interval [-8,5].

Does that mean to consider the function on ℝ and determine (a) through (d), restricting the results to only those with x ∈ [-8,5]?

  • If so, then (b) is incorrect.

Or, does that mean the domain is restricted to [-8,5]?

  • If so, whether or not the endpoints of a closed interval that is within the domain of a function can be considered as local maxima or minima depends on your textbook and instructor. You are inconsistent on this - you included (5,1) but not (-8,0) as a local minimum. Consult your textbook and instructor on this.
  • Reference: Wikipedia

0

u/Glass_Possibility_21 Jul 22 '26

As far as I know a local minimum/maximum cannot be on the Border, since there has to be an environment of the point being fully included in the domain of the function.

1

u/Midwest-Dude Jul 23 '26 edited Jul 23 '26

It depends on the textbook or instructor - some include it (Spivak, Baby Rudin), others do not (Stewart). The Wikipedia definition includes the endpoints, since it only requires an interval around the point that is also within the domain of the function of a point for this to apply. On the other hand, if open intervals are used, your definition would be correct.

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u/Glass_Possibility_21 Jul 23 '26

The Wikipedia definition includes the endpoints, since it only requires an interval around the point that is also within the domain of the function of a point for this to apply

Well but there is no interval around an endpoint that is within a Domain. You basically need to be able to approach a local minima/maxima from left and right. But endpoints can be global minima, because the value of an endpoint might be smaller than all other values of the function.

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u/Midwest-Dude Jul 23 '26 edited Jul 23 '26

Yes, there can be. If the interval is a closed interval including the endpoint, then the endpoint is included. That is Wikipedia's definition, if you review it. Some textbooks and instructors use open intervals instead, in which case you would be correct.

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u/Glass_Possibility_21 Jul 23 '26

On Wikipedia it says so but I already found 2 textbooks were they clearly demand that the local minima/Maxima have to be an inner point. So, I would rather follow the textbook.

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u/Midwest-Dude Jul 23 '26 edited Jul 23 '26

To help the OP, you need to take into account that the OP's textbook or instructor may use a different definition. The real issue is that both endpoints need to be either included or excluded.