How Does the Secant Line Relate to Fermat's Theorem?

[Bipolarity]

I'm trying to understand something in Fermat's Theorem. I can't really phrase it in words, but I will write what my textbook says.

Apparently if

\lim_{x→c}\frac{f(x)-f(c)}{x-c} > 0

then there exists an open interval (a,b) containing c such that

\frac{f(x)-f(c)}{x-c} > 0 for all c in that interval.

How does this follow from the definition of the derivative?

I appreciate all help.

Thanks!

 

[LCKurtz]

That is a general property of limits. If$$
\lim_{x\rightarrow c}g(x) = L > 0$$then there is an open interval $I$ containing $c$ on which $g(x)>0$. It comes directly from the $\epsilon - \delta$ definition of limit.