TITLE: Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$
QUESTION [8 upvotes]: Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$. If $f(0)=0$, find the maximum value of $f(5)$.

$f'(x)=f(x)$ is true when $f(x)=ke^x$. 
$f(x)=0$ satisfies the condition. So $f(5)=0$ which is also the correct answer. But is there any method other than substitution?

REPLY [3 votes]: The inequality indicates
$$
\left|\frac{\mathrm{d}}{\mathrm{d}x}\log\left|f(x)\right|\,\right|\le1
$$
If $\log\left|f(5)\right|=a$ is finite, then $\log\left|f(0)\right|\ge a-5$. Since $f(0)=0$, we must have $f(5)=0$.