Re: On how many arguments the following function can be equal to 0?

On May 9, 6:49 pm, precarion <precar...@xxxxxxxx> wrote:
Hello Everyone! :D

I found yesterday very intersting problem (at least in my opinion) that was presented the previous year in my class, but it was left without an answer and it's now bothering me. If you guys can read the question and give me some hints, I would be very gratefull, 'cause I am not even sure what kind of "mathematical tools" I should use to solve the problem (the Lagrange's mean-value theorem, perhaps?!):

PROBLEM: Check (by providing a proof) for how many arguments x in R the f function can be equal to 0, if we know that: f is differentiable on R and it's such that for all x from R: f'(x) < f(x).

Hope you don't mind me posting such problems on the forum.
Chris

At most once I'd say, because at f(x) = 0 the inequality implies f'(x)
< 0,
so that the next turning point (at which f'(x) = 0) if any, for larger
x
would have f(x) < 0 which contradicts the inequality.


Cheers
John R Ramsden

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