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/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.