Question

If a function is bounded by endpoints and we are asked to find information on absolute/relative extrema or points of inflection, do we consider the endpoints? It's not been clearly clarified but looking at a few practice questions, it seems like it's necessary.

Answer 1

@InsatiableSuffering , you'll probably be the best to refer to since you're in the same class, if you know the answer.
Do we need to consider endpoints when we have to find extrema/inflection points?

Answer 2

Oh and yeah bro this is especially from FRQ perspective (:

Answer 3

Yes, you consider the endpoints as they are part of the function on a closed interval for extrema. The extreme value theorem states that on a closed interval from [a,b], there exists at least one extreme value.

Answer 4

Recall what the point of inflection truly is: it is defined by concavity changing at a point. Since there is no other function beyond endpoints, you can't determine change if the function doesn't exist.

Answer 5

Therefore, you only worry about the function from the open interval (a,b)

Answer 6

Thank you very much for your response, it is greatly appreciated!

So we must list and consider the endpoints. But they are not points of inflection, but they can be extrema due to the extreme value theorem. Is that correct?

Answer 7

Or would they be because their derivative is undefined (for inflection points)?

Answer 8

Yes, you would consider the endpoints for extrema, but not for points of inflection. Points of inflection can only be found on the open interval (a,b), as the endpoints are undifferentiable.

Answer 9

Once again thank you very much for the clarification, it is indeed extremely helpful.

Answer 10

No problem, I've got you.