# Timeless Theorems of Mathematics/Rolle's Theorem

The Rolle\'s theorem says that, If a real-valued function $f$ is
continuous on a closed interval $[a, b]$, differentiable on an open
interval $(a, b)$ and $f(a) = f(b)$, then there exists at least a number
$c$ such that $D_xf(c) = 0$. It means that if a function satisfies the
three conditions mentioned in the previous sentence, then there is at
least a point in the graph of the function, where the slope of the
tangent line at the point is $0$, or the tangent line is parallel to the
$x$-axis.

## Proof

![f(x) is continuous on $[a, b]$ differentiable on $(a, b)$ and
$f(a)=f(b)$. Thus,
$D_xf(c)=0$](RTCalc.svg "f(x) is continuous on [a, b] differentiable on (a, b) and f(a)=f(b). Thus, D_xf(c)=0")

