Prove: $$x \leq f(x) \leq 2x, \forall x\geq0$$
conditions:
$f$ is differentiable
$f(0) = 0$
$1 \leq f'(x) \le 2, \forall x\ge0$
I've tried to do it by limit defn but couldn't seem to get to the right solution:
$$ 1 \le \lim_{x \to c} \frac{f(x)-f(c)}{x-c} \le 2$$
how do i manipulate them in such a way that I get $$x \leq f(x) \leq 2x, \forall x\geq0 $$
I've also noticed that $f(x)$ is an increasing function as $f'(x) > 0$. Is this information of any use?