$f:[0,1]\to R$ is a differentiable function such that f(0) = 0 and  $\left|f^{\mathbf{|}}\left(x\right)\right|\le K\left|f\left(x\right)\right|$ for all  $x\in [0,1](K>0)$  then which of the following is/are always TRUE ?

$$\begin{gathered} {\left(f^{\mathbf{|}}\left(x\right)\right)}^2-K^2{\left(f\left(x\right)\right)}^2\le 0 \\ \Rightarrow (f^{|}\left(x\right)-Kf\left(x\right))(f^{|}\left(x\right)+f\left(x\right))\le 0 \\ \Rightarrow {\left(e^{-kx}f\left(x\right)\right)}^1{\left(e^{kx}f\left(x\right)\right)}^1\le 0 \\ Let g_1\left(x\right)=e^{-kx}f\left(x\right),g_2\left(x\right)=e^{kx}f\left(x\right) \\ \therefore g_1^1\left(x\right).g_2^1\left(x\right)\le 0 \end{gathered}$$

$\therefore$ One of  $g_1\left(x\right),g_2\left(x\right)$ is decreasing
Let  $g_1\left(x\right)$  be increasing but  $g_1\left(0\right)=0\Rightarrow g_1\left(x\right)\ge 0\Rightarrow f\left(x\right)\ge 0$
Let  $g_2\left(x\right)$ be decreasing but  $g_2\left(0\right)=0\Rightarrow g_2\left(x\right)\le 0\Rightarrow f\left(x\right)\le 0$

$\therefore f\left(x\right)=0\forall x\in [0,1]$