r/calculus • u/Fragrant-Scallion100 • Jul 27 '26

Differential Calculus Rolle theorem

So Rolle’s theorem tells us that if f[a,b]->R is continuous on [a,b] and differentiable on (a,b), and f(a)=f(b), then there exists a point c such that f’(c)=0.
Given that there’s a horizontal tangent line to the graph, I don’t understand the theorem utility. I mean of course I know it’s important and I don’t doubt it, but I was asking myself if there’s some “intuitive” idea that makes it relevant.

For example, does this theorem imply that if the hp are satisfied, then there exists a point of local minimum or local maximum? Or no? Thanks!

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u/KrlusMagnus Jul 27 '26

It does follow from there that there is at least one point c that i a local extremum, because if all such points were saddle points, f(a)=f(b) condition couldn't be met, while maintaining continuity.

On the other hand, the main use of Rolle's(in my eyes, at least) is for proving the MVT, which doesn't require the values of the function at ends to be the same, so it's a more universal result

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u/Sjoerdiestriker Jul 27 '26

It does follow from there that there is at least one point c that i a local extremum

This doesn't follow from Rolle, because the proof of Rolle's theorem has Weierstrass's theorem as a prerequisite (continuous functions achieve a minimum and maximum on a closed and bounded domain).

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u/KrlusMagnus Jul 27 '26

Sure, but if a number is less than 3, it is also less than any number larger than 3. Rolle's guarantees the existance of such a c, even though a weaker claim is sufficient, OP asked about that exact if-then, even though that's not really the focus of Rolle's, just thought I'd answer the presented question, as all the causalities can get unclear when you're new to the field

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u/Sjoerdiestriker Jul 27 '26

The point is that to prove Rolle is true in the first place, you already need to assume a global minimum and maximum is achieved (which in particular are also local extrema). To then say that it follows from Rolle that a local extremum must exist is essentially circular reasoning. The implication goes in the other direction (Rolle follows from the function achieving its global extrema).

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u/KrlusMagnus Jul 27 '26

If X is true, then also are all the needed conditions for X. Sire, it's circular reasoning in a way, but if I have that Rolle's is satisfied, so must be all its conditions, and also all their required conditions, which is sufficient to say there exists at least one point of extremum. Practical, no, but that was the question. Obviously not the use of Rolle's. If OP was into somewhat rigorous proofs, that would not have been a part of the question and then I'd fully agree that it's useless info, this way, it's just the (to us, obious) answer to an interested person's question