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Prove that if f(x) = integral from 0 to x of f(t) dt then f = 0

Mathematics
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if \[f(x) = \int\limits_{0}^{x} f(t) dt \] then f = 0
This is not a true statement.
Michael Spivak claims it is

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Other answers:

Who's Michael Spivak?
The man who wrote my textbook.
Whoever he might be, tell him Newton has another opinion :P
This is a direct use of the fundamental theorem of calculus.
\[\int_0^x 2t dt= t^2 \] \[x^2\]
Oh wait!
haha im not going anywhere with this one
It implies f(0)=0.
hes claiming f(x) = 0 for any x
I certinally dont see it
is there derivative sign infront of integral?
Nope that is the whole question
I didn't read the question well at first. This means that f is an anti-derivative of itself, if I'm seeing this right.
yes f'(x) = f(x)
We know that\[f(x)=\int f(x)dx\implies f(x)=e^x\]^^
Yeah.
oh nice
I didnt think about e^x with this one
Then differentiate both sides you get, f'(x)=f(x), which is an ODE that has the solution \(f(x)=ce^{x}\).
Your statement is still not correct :P
Welllll hold on
it is correct....find c
the e^x makes senese with f'(x) = f(x) but this function is an integral
You're saying c=0 @Zarkon.
\[f(0) = \int\limits_{0}^{0} f(t) dt=0\]
It wants me to prove the function is 0
\[\Rightarrow c=0\]
That's right! I'm a loser!! :(
No you're not! So basically because f'(x) = f(x) I can say f(x) = ce^x and then show that f(0) = 0 implying that c=0 so f = 0
Exactly!
makes perfect sense! Im just be a loser now but how do we know there is no other function s.t. f'(x) = f(x)
Mr. Michale Spivak did a good job, that is, to elicit eager students to congregate and think this one through. xd
I hate michale spivak :P
I'm just kidding its just a challening course for me
Lol @across. @Wall, this is a first order homogeneous equation and had only this solution, \(i.e f(x)=ce^{x}\).
homogeneous differential equation* and it has* *_*
ahhh yes! I really like this problem! I can't believe I didnt realize f(x) had to be some form of e^x Thanks for all the help everyone!!!!
You're welcome! Thanks for fanning me :D

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