Lagrange mean value theorem

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Statement

Suppose f is a function defined on a closed interval [a,b] (with a < b) such that the following two conditions hold:

  1. f is a continuous function on the closed interval [a,b] (i.e., it is right continuous at a, left continuous at b, and two-sided continuous at all points in the open interval (a,b)).
  2. f is a differentiable function on the open interval (a,b), i.e., the derivative exists at all points in (a,b). Note that we do not require the derivative of f to be a continuous function.

Then, there exists c in the open interval (a,b) such that the derivative of f at c equals the difference quotient \Delta f(a,b). More explicitly:

f'(c) = \frac{f(b) - f(a)}{b - a}

Geometrically, this is equivalent to stating that the tangent line to the graph of f at c is parallel to the chord joining the points (a,f(a)) and (b,f(b)).

Note that the theorem simply guarantees the existence of c, and does not give a formula for finding such a c (which may or may not be unique).

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