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Secant line in Fermat's Theorem

by Bipolarity
Tags: fermat, line, secant, theorem
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Bipolarity
#1
Mar8-12, 12:07 PM
P: 783
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

[tex] \lim_{x→c}\frac{f(x)-f(c)}{x-c} > 0 [/tex]

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

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

How does this follow from the definition of the derivative?

I appreciate all help.

Thanks!
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LCKurtz
#2
Mar8-12, 02:00 PM
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LCKurtz's Avatar
P: 7,659
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.


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