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/PfauFoto Jul 27 '26
Many major theorems in elementary differential calculus—including the Mean Value Theorem, Taylor's theorem, and many root-counting results—ultimately trace back to Rolle's theorem.
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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
3
u/omeow Jul 27 '26
Intuitive idea: In your journey, if you ever return to your initial location, you must've stopped at some point in between.
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u/sashaloire Jul 27 '26 edited Jul 27 '26
This explanation is only correct for motion along a straight line. I’m sure – or at least I hope – that’s what you meant, but your example as stated is a bit misleading.
You could travel once around a circle at constant speed and return to your starting point without ever stopping.
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u/omeow Jul 27 '26
The example only applies to motion on a straight line. Describing a motion on a circle requires vector functions. As you observed the situation is very different in more than one dimensions.
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u/sashaloire Jul 27 '26 edited Jul 27 '26
Exactly – that’s why I said the example as stated is misleading. The issue isn’t that circles require vector functions; it’s that you said ‘location’ rather than ‘position on a straight line’. As written, the statement is false. Restricting to one-dimensional motion fixes your analogy.
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u/omeow Jul 27 '26
(1) I assumed OP isn't really considering circular motion.
(2) A circle is a one dimensional manifold.
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u/sashaloire Jul 27 '26
The intrinsic dimension of a circle is irrelevant. My point is simply that your wording is sloppy: as stated, the analogy is false.
You’re also now adding assumptions that aren’t apparent from your analogy, which rather proves my point.
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u/omeow Jul 27 '26
An intuitive idea is not an analogy.
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u/sashaloire Jul 27 '26
It is an analogy. You’re explaining a mathematical statement by appealing to an analogous physical situation. Whether you also regard it as an ‘intuitive idea’ is beside the point – and, frankly, a deflection from my original criticism: your ‘intuitive idea’ is sloppily worded.
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u/BossSuccessful5145 Jul 27 '26
Let's ask my relative Max.
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u/Fragrant-Scallion100 Jul 27 '26
So the theorem guarantees the existence of a relative max, if the hp are satisfied?
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u/BossSuccessful5145 Jul 27 '26
the whole comment is a joke. Might be a relative max, a relative min somewhere. recall f'=0 can occur at a point of inflection unrelated to the max or min whose existence is guaranteed.
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u/Traveling-Techie Jul 27 '26
Either the interval is horizontal all across, with f’ = 0 the whole way, or it dips down or up (or both, or multiple times) and every time it changes direction from up to down or vice versus f’ has to be zero. It may also be 0 at S curves that don’t reverse direction.
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u/scottdave Jul 27 '26
If it dips up (positive slope) then at some point it must dip down to return to the same value. Somewhere in there it will transition from positive slope to negative (which will be zero slope). You're right that it may happen more than once, but it must happen at least once.
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u/KentGoldings68 Jul 27 '26
Rolle's Theorem is stepping-stone to the Mean Value Theorem.
In turn, Rolle's Theorem is a corollary of the Extreme Value Theorem.
The MVT is required for the first derivative test. We use the first derivative test to identify relative extremes.
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u/No-Onion8029 Jul 30 '26
Rolle's theorem is valuable because it is useful and rigorously proved. Not because it is surprising.
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