I have a function $f(x)$ continuous on $[0; +\infty)$ and also first and second its derivatives are continuous on the same ray $[0; +\infty)$. So I need to prove:

1) If $f'(x) > 0$ and $f''(x) < 0$ than $f(x)$ is uniformly continuous on $[0;+\infty)$

2) If $f'(x) < 0$ and $f''(x) > 0$ than $f(x)$ is uniformly continuous on $[0;+\infty)$

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> I understand it intuitively because 1) is something like $\sqrt{x}$ which is uniformly continuous on $[0; +\infty)$ and 2) is smth like $\frac{1}{x+1}$. But how to prove it rigorously? $\epsilon-\delta$ language?
>  Would be grateful for any hints.
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