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
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.
