Let $f:\mathbb R\to\mathbb R$ be a function with continuous derivative such that $f(\sqrt{2})=2$ and $$f(x) = \lim_{t\to0}\frac{1}{2t}\int_{x-t}^{x+t} sf'(s)\,ds$$ for all $x\in\mathbb R$. Find $f(3)$.
I guess Fundamental theorem of Calculus needs to be used to solve this.
Taking derivative of x on both sides I simplified the integral to
$(x+t)f'(x+t) - (x-t)f'(x-t) $
The equation becomes:
$f'(x) = \lim (1/2t)(x+t)f'(x+t) - (x-t)f'(x-t) $ as t tends to 0.
This is leading me nowhere. Any ideas on how to tackle this problem?