monotonicity of $\frac{f(x)}{x}$

Question

If $f$ is defined on $[0,\infty)$, and $x_1,x_2>0$, if $\frac{f(x)}{x}$ is monotonic increasing, prove that $f(x_1+x_2)\geq f(x_1)+f(x_2)$.

I suppose it has something to do with starting from $x_1\leq x_1+x_2$ and $x_2\leq x_1+x_2$, but I can't really figure it out-- can anybody help? Thanks!

Answer 1

WLOG we may assume $x_1\geq x_2$. Then we have $$\frac{f(x_1+x_2)}{x_1+x_2}\geq \frac{f(x_1)}{x_1}$$ $$\implies f(x_1+x_2)\geq f(x_1)+x_2\frac{f(x_1)}{x_1}\geq f(x_1)+x_2\frac{f(x_2)}{x_2} = f(x_1)+f(x_2)$$which is what we wanted to show.