I am stuck at the following exercise:
Let $f:\left[0, \infty \right) \rightarrow\mathbb{R}$ be a continuous and differentiable function. We make the following assumptions:
$f(0) = 0 $
$f'(0) = -1$
$\exists K > 0 \text{ such that } \left| f'(t_2)-f'(t_1)\right| \le K\left|t_2-t_1\right| (\forall t_2,t_1 > 0)$
Prove the following points:
Find some coefficients $a>0$ and $b<0$ such that $f(t) \le at^2+bt \ \ (\forall t \ge 0)$
Let $s \in \left(0,1\right) \text{ and } t_0 = \frac{2(1-s)}{K} > 0$. Show that for all $t \in \left[0,t_0\right]$ the Armijo rule $f(t) \le -st$ holds.
Thanks to the help of achille hui I got 1) now; we can choose $a = K/2$ and $b=-1$. For 2) I think the upper bound obtained from 1) would be useful:
$$f(x) \le \frac{K}{2}t^2 - t $$
I tried to insert the $t$'s from the interval $\left[0,t_0\right]$ but it did not give me any insights. Could you give me a hint?