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

Question

anonymous · Answer

if $$f(x) = \int\limits_{0}^{x} f(t) dt $$ 
then f = 0

mr.math · Answer

This is not a true statement.

anonymous · Answer

Michael Spivak claims it is

mr.math · Answer

Who's Michael Spivak?

anonymous · Answer

The man who wrote my textbook.

mr.math · Answer

Whoever he might be, tell him Newton has another opinion :P

mr.math · Answer

This is a direct use of the fundamental theorem of calculus.

anonymous · Answer

$$\int_0^x 2t dt=  t^2 $$

$$x^2$$

mr.math · Answer

Oh wait!

anonymous · Answer

haha im not going anywhere with this one

across · Answer

It implies f(0)=0.

anonymous · Answer

hes claiming f(x) = 0 for any x

anonymous · Answer

I certinally dont see it

anonymous · Answer

is there derivative sign infront of integral?

anonymous · Answer

Nope that is the whole question

mr.math · Answer

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.

anonymous · Answer

yes f'(x) = f(x)

across · Answer

We know that$$f(x)=\int f(x)dx\implies f(x)=e^x$$^^

mr.math · Answer

Yeah.

anonymous · Answer

oh nice

anonymous · Answer

I didnt think about e^x with this one

mr.math · Answer

Then differentiate both sides you get, f'(x)=f(x), which is an ODE that has the solution $f(x)=ce^{x}$.

mr.math · Answer

Your statement is still not correct :P

anonymous · Answer

Welllll hold on

zarkon · Answer

it is correct....find c

anonymous · Answer

the e^x makes senese with f'(x) = f(x) but this function is an integral

mr.math · Answer

You're saying c=0 @Zarkon.

zarkon · Answer

$$f(0) = \int\limits_{0}^{0} f(t) dt=0$$

anonymous · Answer

It wants me to prove the function is 0

zarkon · Answer

$$\Rightarrow c=0$$

mr.math · Answer

That's right! I'm a loser!! :(

anonymous · Answer

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

mr.math · Answer

Exactly!

anonymous · Answer

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)

across · Answer

Mr. Michale Spivak did a good job, that is, to elicit eager students to congregate and think this one through. xd

anonymous · Answer

I hate michale spivak :P

anonymous · Answer

I'm just kidding its just a challening course for me

mr.math · Answer

Lol @across. 
@Wall, this is a first order homogeneous equation and had only this solution, $i.e f(x)=ce^{x}$.

mr.math · Answer

homogeneous differential equation* and it has* *_*

anonymous · Answer

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!!!!

mr.math · Answer

You're welcome! Thanks for fanning me :D