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In mathematics, the mean value theorem states, roughly: that given a planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval. More precisely, if a function ''f'' is continuous on the closed interval (''b'' ), where ''a'' < ''b'', and differentiable on the open interval (''a'', ''b''), then there exists a point ''c'' in (''a'', ''b'') such that : A special case of this theorem was first described by Parameshvara (1370–1460), from the Kerala school of astronomy and mathematics in India, in his commentaries on Govindasvāmi and Bhaskara II.〔J. J. O'Connor and E. F. Robertson (2000). (Paramesvara ), ''MacTutor History of Mathematics archive''.〕 The mean value theorem in its modern form was later stated by Augustin Louis Cauchy (1789–1857). It is one of the most important results in differential calculus, as well as one of the most important theorems in mathematical analysis, and is useful in proving the fundamental theorem of calculus. The mean value theorem follows from the more specific statement of Rolle's theorem, and can be used to prove the more general statement of Taylor's theorem (with Lagrange form of the remainder term). == Formal statement == Let ''f'' : (''b'' ) → R be a continuous function on the closed interval (''b'' ), and differentiable on the open interval (''a'', ''b''), where Then there exists some ''c'' in (''a'', ''b'') such that :: The mean value theorem is a generalization of Rolle's theorem, which assumes ''f''(''a'') = ''f''(''b''), so that the right-hand side above is zero. The mean value theorem is still valid in a slightly more general setting. One only needs to assume that ''f'' : (''b'' ) → R is continuous on (''b'' ), and that for every ''x'' in (''a'', ''b'') the limit : exists as a finite number or equals +∞ or −∞. If finite, that limit equals ''f′''(''x''). An example where this version of the theorem applies is given by the real-valued cube root function mapping ''x'' to ''x''1/3, whose derivative tends to infinity at the origin. Note that the theorem, as stated, is false if a differentiable function is complex-valued instead of real-valued. For example, define for all real ''x''. Then :''f''(2π) − ''f''(0) = 0 = 0(2π − 0) while ''f′''(''x'') ≠ 0 for any real ''x''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mean value theorem」の詳細全文を読む スポンサード リンク
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