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i need to prove:

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if a differentiable function f:(a,b) ----> R (reals) assumes a max or a min at some x element of (a,b), prove that f'(x) = 0. why is this assertion false when [a,b] replaces (a,b)?

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-I'm stumped at where to start...because if i knew that f was continuous on [a,b] and f(a) = f(b), this would be Rolle's thm...which i think i could prove...

-or if f(a) did not equal f(b), then maybe i could get somewhere by mean value thm... (that is, if f was cont on [a,b])

-BUT i dont know that f is cont on [a,b]...

-the only thing i know given what I have is that since f is differentiable on (a,b) then its derivative f'(x) has the intermediate value property....can this get me anywhere?

-any help on how to get started on this proof would be greatly appreciated! thanks for reading through my lengthy post! :)

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# Analysis- if f assumes max/min for x in (a,b) prove f'(x) = 0

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