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?