Question 97
Let $f: \mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ be a function such that for all $a, b \in \mathbb{R}^{\ge 0}$:
i) $f(a) = 0 \Leftrightarrow a = 0$.
ii) $f(ab) = f(a)f(b)$.
iii) $f(a + b) \le 2 \max\{f(a), f(b)\}$.
Prove that for every $a,b \in \mathbb{R}^{\ge 0}$, $f(a+b) \le f(a) + f(b)$.
i) $f(a) = 0 \Leftrightarrow a = 0$.
ii) $f(ab) = f(a)f(b)$.
iii) $f(a + b) \le 2 \max\{f(a), f(b)\}$.
Prove that for every $a,b \in \mathbb{R}^{\ge 0}$, $f(a+b) \le f(a) + f(b)$.