Question 111
Suppose that a function $f: [0, \infty) \to [0, \infty)$ satisfies
(1) for all $x, y \ge 0$, $f(x)f(y) \le y^2 f\left(\frac{x}{2}\right) + x^2 f\left(\frac{y}{2}\right)$;
(2) for all $0 \le x \le 1$, $f(x) \le 2016$.
Prove that $f(x) \le x^2$ for all $x \ge 0$.
(1) for all $x, y \ge 0$, $f(x)f(y) \le y^2 f\left(\frac{x}{2}\right) + x^2 f\left(\frac{y}{2}\right)$;
(2) for all $0 \le x \le 1$, $f(x) \le 2016$.
Prove that $f(x) \le x^2$ for all $x \ge 0$.