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Is this an abuse of Rolle's theorem?

Rolle's theorem

If a real-valued functionfis continuous on a proper closed interval [a,b], differentiable on the open interval (a,b), andf(a) =f(b), then there exists at least onecin the open interval (a,b) such thatf'(c) = 0.

##[x_1, x_1]##, which means ##x_1\leq x\leq x_1##, is not an interval but a point. And ##(x_1, x_1)##, which means ##x_1<x<x_1##, doesn't make sense. So how can we apply Rolle's theorem?

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# I Use Rolle's theorem to show repeated root has zero gradient

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