If $f(x)=\int_a^x f(t)dt$, is $f(x)=0$?

Question

How can I prove this?

Knowing $f:[a,b] \rightarrow \mathbb{R}$ is continuos in $[a,b]$ and $f(x)=\displaystyle\int _a^x f(t)dt$,

how can I conclude that $f(x)=0$ $\forall x \in [a,b]$? I can't even find the first step

Answer 1

This just mean that $f$ is a continuous function verifying the two conditions:

$f’(x)=f(x){\tag 1}$

$f(a)=0{\tag 2}$

The first condition gives: $ f(x)=ke^x$ and since $f(a)=ke^a=0$ then $k=0$.

The answer is therefore yes!

Answer 2

$f'(x) = f(x)$ by the fundamental theorem of calculus. Also by the same theorem this derivative exists.

This leads to the ODE with solution $f(x) = Ae^x$.

Also $f(a) = 0$. We must conclude that $A=0$.

Sorted