Is square root sublinear?

Question

I want to prove that $\sqrt x $ is sublinear, but i don't know how to do it, precisely i'm having difficulties stating that $\sqrt x $ is sublinear in $[0, 1]$ $$\\ f(x)\ is\ sublinear\ \iff\ \exists\ a,b \in\ \mathbb R\ \mid\ |f(x)| \le a|x| + b $$

Answer 1

Your $x$ is non-negative. We need $a$ and $b$ such that

$$ \forall x \ge 0 \quad 0\le\sqrt x \le ax +b.$$

Raise to the power $2$: $$0 \le a^2 x^2 + (2 a b - 1) x + b^2$$

Take for simplicity $a=1$: $$0 \le x^2 + (2 b - 1) x + b^2,$$ and that for $x\ge0$. It is sufficient to take $2b-1\ge0$.