Rolle's Theorem
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Rolle's Theorem

Theorem 1 (Rolle's Theorem): If $f$ is a function that satisfies the following conditions:
a) $f$ is a continuous function over the closed interval $[a, b]$.
b) $f$ is differentiable on the open interval $(a, b)$.
c) $f(a) = f(b)$.
Then there exists a value $c \in (a, b)$ such that $f'(c) = 0$.

Essentially, Rolle's theorem says that a function always has a point $c$ where $f(a) = f(b)$, $f'(c) = 0$ on an interval $(a, b)$ given it is continuous on $[a, b]$ and differentiable on $(a, b)$ - that is, to connect two points $(a, f(a))$ and $(b, f(b))$, there is a point $c$ that has a horizontal slope.

Clearly in the case where the function $f$ is a line there is a horizontal tangent everywhere since the entire function is a horizontal line. In the case where the function $f$ is not simply a horizontal line, then there a value c such that on the interval $(a, c)$, $f$ is either increasing or decreasing. There must be another interval oppositely increasing/decreasing which creates a maximum or minimum value. Thus there will always be a horizontal tangent under the three conditions above.

Rolle's Theorem is a special case of the more generalized Mean Value Theorem.

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