Question 15
a) Consider $f: [0, \infty) \to [0, \infty)$ a differentiable and convex function such that $f(x) \le x$, for $x \ge 0$. Then $f'(x) \le 1$, for all $x \ge 0$.
b) Determine all differentiable and convex functions $f: [0, \infty) \to [0, \infty)$ with the properties $f(0) = 0$ and $f'(x) \cdot f(f(x)) = x$, for all $x \ge 0$.
b) Determine all differentiable and convex functions $f: [0, \infty) \to [0, \infty)$ with the properties $f(0) = 0$ and $f'(x) \cdot f(f(x)) = x$, for all $x \ge 0$.