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Zero function

July 7, 2018May 2, 2019 Tolaso

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(0)=0$ and $\left| f'(x) \right| \leq |f(x)|$ forall $x \in \mathbb{R}$ . Prove that $f(x)=0$ forall $x \in \mathbb{R}$ .

Solution

We note that

$\begin{align*} \left ( e^{-2x} f^2(x)\right ) ' &= 2e^{-2x}\left ( f(x)f'(x) - f^2(x) \right ) \\ &\leq 2e^{-2x}\left ( |f(x)f'(x)| - f^2(x) \right ) \\ &= 2e^{-2x}|f(x)|\left ( |f'(x)| - |f(x)| \right ) \le 0 \end{align*}$

Hence $e^{-2x} f^2(x)$ is decreasing. Therefore , $e^{-2x} f^2(x) \leq 1 \cdot f^2(0) =0$ . The result follows.

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