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 $D_{x}f(c)=0$. 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, $D_{x}f(c)=0$