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

- anonymous

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

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- anonymous

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

- Mr.Math

This is not a true statement.

- anonymous

Michael Spivak claims it is

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## More answers

- Mr.Math

Who's Michael Spivak?

- anonymous

The man who wrote my textbook.

- Mr.Math

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

- Mr.Math

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

- anonymous

\[\int_0^x 2t dt= t^2 \]
\[x^2\]

- Mr.Math

Oh wait!

- anonymous

haha im not going anywhere with this one

- across

It implies f(0)=0.

- anonymous

hes claiming f(x) = 0 for any x

- anonymous

I certinally dont see it

- anonymous

is there derivative sign infront of integral?

- anonymous

Nope that is the whole question

- Mr.Math

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

yes f'(x) = f(x)

- across

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

- Mr.Math

Yeah.

- anonymous

oh nice

- anonymous

I didnt think about e^x with this one

- Mr.Math

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

- Mr.Math

Your statement is still not correct :P

- anonymous

Welllll hold on

- Zarkon

it is correct....find c

- anonymous

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

- Mr.Math

You're saying c=0 @Zarkon.

- Zarkon

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

- anonymous

It wants me to prove the function is 0

- Zarkon

\[\Rightarrow c=0\]

- Mr.Math

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

- anonymous

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

Exactly!

- anonymous

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

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

- anonymous

I hate michale spivak :P

- anonymous

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

- Mr.Math

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

- Mr.Math

homogeneous differential equation* and it has* *_*

- anonymous

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

You're welcome! Thanks for fanning me :D

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