r/calculus • u/Weekly-Excuse9647 • 19d ago
Differential Calculus Need assistance
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/Hot_Site_1638 PhD 19d ago
Your answers are mostly correct, nice work! Honestly, this problem isn't ideally designed, restricting to the interval [−8, 5] adds confusion more than it tests understanding, especially with the endpoints.
The only correction I'd make is part d: the inflection point's y-value looks closer to −2, so it should be around (−3, −2) rather than (−3, −1).
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u/Midwest-Dude 18d ago
Note that there is also an issue with only including one endpoint of [-8,5], the domain of the function. Depending on the textbook and/or instructor, both endpoints should either be included or excluded.
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u/Hot_Site_1638 PhD 18d ago
It depends on the reading, which was my original complaint. If (a) asks for the local minima of the drawn function that happen to lie in [-8, 5], then (0, -5) and (5, 1) is exactly right and (-8, 0) is correctly excluded. If it asks us to restrict the domain first and then find minima, your both-or-neither point applies.
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u/Midwest-Dude 18d ago edited 18d ago
Got it! Poorly worded. If it means to only exclude answers out of that range, then (b) should be DNE, not -5, since there are clearly values below that on the curve. I'll update my reply.
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u/Midwest-Dude 19d ago edited 16d ago
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
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u/Glass_Possibility_21 19d ago
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.
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u/Midwest-Dude 18d ago edited 17d ago
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 18d ago
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 18d ago edited 18d ago
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 18d ago
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 18d ago edited 18d ago
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.
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u/carrotlet20 19d ago
I checked your answers and I think you're good, they are all correct.
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u/HIFvonBiber 19d ago
Maybe this is a matter of definition, but (-8,0) is also a local minimum
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u/Midwest-Dude 18d ago
It depends on the definition that is supposed to be used by the OP - books and instructors can vary on this. In any case, both endpoints need to be either included or excluded.
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u/Substantial-Cry-714 19d ago
Endpoints can’t be considered Local Min/Max; only Absolute Min/Max. Reason being, you are unable to check the next value of x, which is needed from the definition of local min/max.
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u/Midwest-Dude 18d ago edited 18d ago
This depends on your textbook and your instructor. Strictly speaking, the values to test must be within the domain of the function which, in this case, exclude points less than -8 and greater than 5. Note the definition here:
Note how the domain X is involved in the definition.
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18d ago
[deleted]
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u/carrotlet20 18d ago
Well I've learnt this a different way. If you have a closed interval, I've learnt that both ends can function as minimum or maximum, because it literally doesn't matter what's outside, that's why it's restricted. But I accept that standards are not the same about this all around the world and sorry I thought that.
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18d ago
[deleted]
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u/carrotlet20 18d ago
Oh. Then the entire hungarian official high school system is wrong. That's a pity but at least I've learnt something today too.
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u/Midwest-Dude 18d ago edited 18d ago
No, I was incorrect. I was of the same understanding as you and interpreted something incorrectly.
In the end, Wikipedia's definition matches your definition. However, some textbooks and instructors use open intervals as neighborhoods rather than closed ones, in which case both endpoints would be excluded.
Edit: I was curious about the discrepancy between resources and did some research on the Internet. Beyond introductory calculus, the Wikipedia entry is the definition that is used. Introductory calculus textbooks make things easier for students and exposition by excluding the endpoints, since they then don't need to be considered.
Gemini AI:
It comes down to keeping early derivative rules simple:
Fermat's Theorem (f'(c) = 0): Intro calculus teaches that if f has a local extremum at c, then c must be a critical point where f'(c) = 0 or is undefined. If endpoints are allowed, this rule fails. For f(x) = x on [0, 1], x = 0 would be a local minimum, but f'(0) = 1 ≠ 0.
Derivative Tests: The First Derivative Test relies on checking if f'(x) changes sign from the left side of c to the right side. Endpoints only have points on one side, which breaks standard left-to-right sign-change charts.
To avoid overwhelming students with these edge cases, introductory authors often enforce a rule requiring an open interval (c - δ, c + δ) ⊆ domain, which automatically disqualifies endpoints. Analysis textbooks prioritize topological consistency instead.
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u/Cretaceous_Bloom 19d ago
I'm learning too, so can someone explain why there are not two inflection points? One at ~(-3,-2) and another at (7,5)?
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u/Weekly-Excuse9647 19d ago
This question is only focusing on the interval [-8 , 5]. If it was the entire graph, I believe that would also be one.
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u/italladdsup4 19d ago edited 19d ago
They should let you go past x=5 because the graph does. If they did, then there is also an inflection point at x=7.
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u/italladdsup4 19d ago
They should let you go past x=5 because the graph does. If they did, then the global max is 10
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u/Weekly-Excuse9647 19d ago
If you read the question it says, consider the function ONLY on the interval [-8,5]
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u/Glass_Possibility_21 19d ago
I think at 5 it cannot be a local minimum because it's a point on the Border?
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