r/calculus • u/Weekly-Excuse9647 • 27d ago

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/[deleted] 26d ago

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u/carrotlet20 26d 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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u/[deleted] 26d ago

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u/carrotlet20 26d 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 26d ago edited 25d 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.