Question

Let $f(x)$ be a non-negative differentiable function on $[0, \infty)$ such that $f (0)=0$ and $f'(x) \le 2f(x)$ for all $x>\,0$ Then, on $[0, \infty)$.

• A. $f (x)$ is always a constant function
• B. $f (x)$ is strictly increasing
• C. $f (x)$ is strictly decreasing
• D. $f'(x)$ changes sign

Answer 1

Answer: (A)

We have,
$f (x)$ is non-negative differentiable function on $[0, \infty)$
$f(0)=0, f' (x) \le 2f (x)$
$\Rightarrow f'(x) \le 2 f (x)$
$\Rightarrow \log f (x) \le 2x+c$
$\Rightarrow f (x) \le Ae^{2x}$
$\Rightarrow f (0) \le A$
$\Rightarrow A=0$
$[f (0)=0$ and $f (x) \le 0]$
$\therefore f (x)=0$
Hence, $f (x)$ is always a constant function