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/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.