# Rolle's theorem and open interval

• July 11th 2012, 01:55 PM
loui1410
Rolle's theorem and open interval
Prove: if $f(x)$ is differentiable on the open interval $(a,b)$ and $\lim_{x\to a+}f(x) = \lim_{x\to b-}f(x)$ then there exists $c$ in the interval such that $f'(c)=0$.

It's almost clear that I need to use Rolle's theorem, but how do I overcome the fact that the given $(a,b)$ is an open interval?
• July 11th 2012, 03:02 PM
Reckoner
Re: Rolle's theorem and open interval
Quote:

Originally Posted by loui1410
Prove: if $f(x)$ is differentiable on the open interval $(a,b)$ and $\lim_{x\to a+}f(x) = \lim_{x\to b-}f(x)$ then there exists $c$ in the interval such that $f'(c)=0$.

Let

$L = \lim_{x\to a^+}f(x) = \lim_{x\to b^-}f(x).$

Define a new function $g(x)$ as follows:

$g(x) = \left\{\begin{array}{ll}L,&x = a\\f(x),&a < x < b\\L,&x = b\end{array}\right.$

It should be easy to show that $g$ is continuous on $[a, b]$ and differentiable on $(a, b),$ and since $g(a) = g(b),$ the requirements for Rolle's theorem are satisfied. And since $f'(x) = g'(x)$ for $a $g'(c) = 0\Rightarrow f'(c) = 0.$