Question

Why does a function being well defines and it having a point c where lim_|x->c|{((f(x)-f(c))/(x-c)^2}=A < 0 guarantee that f'(c) exists and that c is a maximum point?

Answer 1

I believe I understand the 2nd part of it. Since the bottom of the limit is positive at all times, and the result is negative, that means that f(x)-f(c) is negative on both sides of the limit and use that to prove that f(c) is a maximum point. What I'm really stuck on is proving differentiability. I'll repost the limit with symbols:\[\lim_{x \rightarrow c}(f(x)-f(c))/(x-c)^2 = A < 0\]