Question 24
a) Let $f:[0,\infty)\to[0,\infty)$ be differentiable and convex. Show that if $f(x)\le x$ for every $x\ge0$, then $f'(x)\le1$ for every $x\ge0$.
b) Determine the differentiable convex functions $f:[0,\infty)\to[0,\infty)$ satisfying $f(0)=0$ and $f'(x)f(f(x))=x$ for every $x\ge0$.
Problem (the source repeats the statement in Hungarian):
a) The function $f:[0,\infty)\to[0,\infty)$ is differentiable and convex. Prove that if $f(x)\le x$ for every $x\ge0$, then $f'(x)\le1$ for every $x\ge0$.
b) Determine all differentiable convex functions $f:[0,\infty)\to[0,\infty)$ for which $f(0)=0$ and $f'(x)f(f(x))=x$ for every $x\ge0$.
b) Determine the differentiable convex functions $f:[0,\infty)\to[0,\infty)$ satisfying $f(0)=0$ and $f'(x)f(f(x))=x$ for every $x\ge0$.
Problem (the source repeats the statement in Hungarian):
a) The function $f:[0,\infty)\to[0,\infty)$ is differentiable and convex. Prove that if $f(x)\le x$ for every $x\ge0$, then $f'(x)\le1$ for every $x\ge0$.
b) Determine all differentiable convex functions $f:[0,\infty)\to[0,\infty)$ for which $f(0)=0$ and $f'(x)f(f(x))=x$ for every $x\ge0$.