Rolle's Theorem

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 x₀, a <>x₀<>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

Figure 1

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 alsoDIFFERENTIAL CALCULUS.)

单词	rolle's theorem
释义	Rolle's theorem Rolle's theorem R0289300 (rôlz, rōlz)n. A theorem stating that if a curve is continuous, has two x-intercepts, and has a tangent at every point between the intercepts, at least one of these tangents is parallel to the x-axis. [After Michel Rolle (1652-1719), French mathematician.] Rolle's Theorem 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 x₀, a <>x₀<>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 Figure 1 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 alsoDIFFERENTIAL CALCULUS.)
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