Prove each of the following conditions is sufficient to ensure that $f(x+y)≤f(x)+f(y)$ $f$ is increasing and $f$ satisfies $f(x)=0$ and $f(x)>0$ for all $x>0$.
Show that each of the following conditions is sufficient to ensure that $f(x+y)≤f(x)+f(y)$ for all $x,y≥0$.


*

*(a) $f$ has a second derivative satisfying $f''＜0$;

*(b) $f$ has a decreasing first derivative;

*(c) $\frac{f(x)}{x}$ is decreasing for $x>0$;


I know $a → b → c$.
I don't know how to prove $f(x+y)≤f(x)+f(y)$ using (c).

Hint: Assume (c). Then
$$x\frac{f(x+y)}{x+y}\le x\frac{f(x)}{x} $$
and
$$y\frac{f(x+y)}{x+y}\le y\frac{f(y)}{y}.$$
