Recently I've come across the following task:
Suppose $f(x)$ is defined and continuous on $[0, + \infty)$
$f(x) > 0, f'(x) > 0 \ \forall \ x \in [0, + \infty)$
$f(0) = e$
$\forall \ x >0 \ f(x) > f'(x)$ Then
A) $f(1) < e$
B) $f(e) < e^e$
C) $f(2e) < e^8$
D) $f(e^2) < e^{e^2}$
E) A-D are false
What I've done: assume $f(x) = e^x + e - 1$ then $f(0) = e$, $f'(x) = e^x < e^x + e - 1$, so this function satisfies all conditions hence A), B) and D) are false. But I have no idea, how to prove that C) is correct. What should I use? Something like MVT?
Could you please help me?
Thanks a lot in advance!
$f'(x)<f(x)$ and $f(0)=e$ implies that $f(x)<e^{x+1}$ for positive values of $x$. I think that should be enough to figure out everything in the question.