# Rolle's Theorem

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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 exists*x*_{0},*a*<>*x*_{0}<>*b*, such that ƒ′(*x*^{0})=0.## Rolle’s Theorem

a theorem of mathematical analysis first stated by M. Rolle in 1690. According to this theorem, if the function *f(x*) is continuous on the closed interval *[a, b]*, has a definite derivative within the interval, and takes on the equal values *f(a) = f(b*) at the ends of the interval, then the function’s

derivative *f’(x)* vanishes at least once in the interval (*a, b)*—that is, there exists a *c, a < c < b*, such that *f’(c)* = 0. A corollary of the theorem provides that the derivative of a function has at least one zero between two successive zeros of the function. Geometrically, Rolle’s theorem is self-evident (see Figure 1). (*See also*DIFFERENTIAL CALCULUS.)