Rolle's Theorem Proof

The theorem states that equal values at two distinct points should have a point at someplace between them, where the first derived item will be zero i.e. the slope of the tangent line to the graph of the function.

Rolle's Theorem Definition, Proof

Statement
    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 are parallel to the x-axis.

Diagram
   
Rolle Theorem


To Prove
    If f(x) is continuous an [a, b] and differentiable on (a, b) and if f(a) = f(b) then there is some c in the interval (a,   b) such that f'(c) = 0.
     f(a) = f(b)

Proof
    The proof makes use of the mathematical induction.

     For n = 1 is a simply standard edition of the Rolle's Theorem.

     As induction hypothesis, presume the generalization is true for n - 1. We need to prove it for n > 1.

     With the available standard version of the Rolle's Theorem definition, for every integer k from 1 to n, there is a ck
     accessible in the open interval (ak, bk) such that f' (ck) = 0. Hence the first derivative pleases the
     hypothesis with the n - 1 closed intervals [c1, c2], [cn - 1, cn].

     By the introduction of the hypothesis and also the f is differentiable at c, so the left and right limits must occur
     during the same time, so there is a 'c' such that the (n - 1)st derivative of f' at c is zero.


     

Hence the Rolle's theorem is proved.

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