Rolle's theorem

Rolle's theorem

Special case of the mean-value theorem of differential calculus. It states that if a continuous curve passes through the x-axis twice within a given interval and has a unique tangent line at every point of that interval, then somewhere between the two points of interception it has a tangent parallel to the x-axis.

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Rolle's theorem [′rȯlz ‚thir·əm]

(mathematics)

If a function ƒ(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) and if ƒ(a) = ƒ(b), then there existsx₀,a<x₀<b, such that ƒ′(x⁰)=0.