Question 134
We consider the functions $f: \mathbb{R} \to \mathbb{R}$ for which:
$f(x+y)f(x-y) \geq (f(x))^2 - (f(y))^2$, for all $x, y \in \mathbb{R}$.
We suppose that there exists $x_0, y_0 \in \mathbb{R}$ such that
$$
f(x_0 + y_0)f(x_0 - y_0) \gt (f(x_0))^2 - (f(y_0))^2.
$$
Prove that $f(x) \geq 0$, for every $x \in \mathbb{R}$ or $f(x) \leq 0$, for every $x \in \mathbb{R}$.
$f(x+y)f(x-y) \geq (f(x))^2 - (f(y))^2$, for all $x, y \in \mathbb{R}$.
We suppose that there exists $x_0, y_0 \in \mathbb{R}$ such that
$$
f(x_0 + y_0)f(x_0 - y_0) \gt (f(x_0))^2 - (f(y_0))^2.
$$
Prove that $f(x) \geq 0$, for every $x \in \mathbb{R}$ or $f(x) \leq 0$, for every $x \in \mathbb{R}$.