Question 77
Find all differentiable functions $f: \mathbb{R} \to \mathbb{R}$, whose derivative is bounded in a neighborhood of the origin and fulfill the condition
$$
x f(x) - y f(y) = (x^2 - y^2) \max(f'(x), f'(y)),
$$
for every real numbers $x$ and $y$.
$$
x f(x) - y f(y) = (x^2 - y^2) \max(f'(x), f'(y)),
$$
for every real numbers $x$ and $y$.