TITLE: For nonnegative continuous $f$, if $f'(x)-f(x)\leq 0, \forall x\geq 0$ and $f(0)=0$, find the value of $f(1)$.
QUESTION [5 upvotes]: Let $f(x)$ be a non-negative continuous function such that $f'(x)-f(x)\leq 0, \forall x\geq 0$ and $f(0)=0,$find the value of $f(1)$.

$f'(x)-f(x)\leq 0$$\Rightarrow f'(x)\leq f(x)$$\Rightarrow \frac{f'(x)}{f(x)}\leq 1$$\Rightarrow \int\frac{f'(x)}{f(x)}dx\leq \int1 dx$$\Rightarrow \log f(x) \leq x$.Then could not solve further.Can someone assist me find $f(1)$?

REPLY [5 votes]: \begin{align*}
f(1)
&= e\left[ e^{-1} f(1) \right] \\
&= e\left[ e^0 f(0) + \int_0^1 \frac{d}{dx} (e^{-x} f(x)) \; dx \right] \\
&= e\left[ 0 + \int_0^1 e^{-x} \left[ f'(x) - f(x) \right] \right] \\
&\le e\left[ \int_0^1 e^{-x} \left[ 0 \right] \right] \\
&= 0.
\end{align*}