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Mean value theorem
On the existence of a tangent to an arc parallel to the line through its endpoints / From Wikipedia, the free encyclopedia
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In mathematics, the mean value theorem (or Lagrange 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.
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More precisely, the theorem states that if is a continuous function on the closed interval
and differentiable on the open interval
, then there exists a point
in
such that the tangent at
is parallel to the secant line through the endpoints
and
, that is,