Let $x(t)\in C^1(\mathbb{R}_+,\mathbb{R}^m)$ be a vector-function such that

1. $\|x(t)\|+\|\dot x(t)\|\to 0$ as $t\to\infty$ and
2. for all $t>0$ one has $\|x(t)\|\le c_1\|\dot x(t)\|$

Is it true that $\|x(t)\|\le c_2 e^{-c_3t}$? Here $c_i$ are positive constants.

Let $x(t)\in C^1(\mathbb{R}_+,\mathbb{R}^m)$ be a vector-function such that

1. $\|x(t)\|+\|\dot x(t)\|\to 0$ as $t\to\infty$ and
2. for some real $c_1>0$ and all $t>0$ one has $\|x(t)\|\le c_1\|\dot x(t)\|$

Is it true that $\|x(t)\|\le c_2 e^{-c_3t}$ all $t>0$? Here $c_i$ are positive constants.