For what values of $$s\in [0,\infty),$$ the function given by $$f(x)=x^s$$ sub-additive?
i.e. $$f(a+b)\leq f(a)+f(b) \quad \forall a,b \in [0, \infty)$$
I feel $$f$$ is sub-additive for $$s\in [0,1]$$ and not sub-additive for $$s>1$$
For $$s >1$$ just take $$a=b=1$$ to see that it is not subadditive. For $$s<1$$ consider the function $$f(a)=a^{s}+b^{s}-(a+b)^{s}$$ with $$b$$ fixed. Note that $$f'(a)=sa^{s-1}-s(a+b)^{s-1}$$. Since $$s-1 <0$$ this shows that $$f'(a)>0$$ so $$f$$ is increasing. Also $$f(0)=0$$ so $$f(a) \geq 0$$ for all $$a \geq 0$$.