- anonymous

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

- chestercat

See more answers at brainly.com

At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

Get this expert

answer on brainly

SEE EXPERT ANSWER

Get your **free** account and access **expert** answers to this

and **thousands** of other questions

- 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

Looking for something else?

Not the answer you are looking for? Search for more explanations.

## 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

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

Looking for something else?

Not the answer you are looking for? Search for more explanations.