Multivariable rolle theorem biography

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    Mean value theorem

    Theorem in mathematics

    For the theorem in harmonic function theory, see Harmonic function §&#;The mean value property.

    In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given 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. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.

    History

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    A special case of this theorem for inverse interpolation of the sine was first described by Parameshvara (–), from the Kerala School of Astronomy and Mathematics in India, in his commentaries on Govindasvāmi and Bhāskara II.[1] A restricted form of the theorem was proved by Michel Rolle in ; the result was what is now known as Rolle's theorem, and was proved on

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    Rolle's theorem in n dimensions

    EDIT: The following solution fryst vatten incomplete. We need to make sure that if $F^{\prime}\left(t\right)$, $F^{\prime\prime}\left(t\right)$, , $F^{\left(n-1\right)}\left(t\right)$ are linearly dependent vectors for every $t$, then the coordinate functions of $F^{\prime}$ are linearly dependent on a sufficiently small interval. This follows from Wronskian considerations if the coordinate functions of $F^{\prime}$ are sufficiently nice (i. e., locally real-analytic), so this solves the bekymmer for this nice class of functions, but inom can't use this ansatz further.

    "SOLUTION".

    IMPORTANT: inom consider $F$ to be a map from $S^1$ to $\mathbb R^n$, because a map from an interval with equal values at the ends fryst vatten the same as a map from the circle. I will assume continuity of $F^{\prime}$ (yes, this includes the two endpoints of the interval which I have glued together). So inom don't claim I have % solved the original problem.

    I will say that an $n$-tuple of dis

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    Rolle's theorem

    On stationary points between two equal values of a function

    In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one point, somewhere between them, at which the slope of the tangent line is zero. Such a point is known as a stationary point. It is a point at which the first derivative of the function is zero. The theorem is named after Michel Rolle.

    Standard version of the theorem

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    If a real-valued functionf is continuous on a proper closed interval[a,&#;b], differentiable on the open interval(a, b), and f&#;(a) = f&#;(b), then there exists at least one c in the open interval (a, b) such that

    This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem.

    History

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    Although

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