Does this inequality hold true for all $\alpha\in\mathbb{R}$? Let $x$ and $y$ be two real and positive numbers. Let $\alpha\in\mathbb{R}$. I am trying to understand if the inequality
$$ x^{\alpha} + y^{\alpha} \leq (x+y)^{\alpha}$$
holds true. By attemps, I found that it holds true only if $\alpha \geq1$. Could anyone please tell me if it is true?
Moreover, there is a simple way to justify it instead of proceeding by trial?
Thank you in advance!

If the  inequality holds for all $x, y \geq 0$ the take $x=y=1$ to get $2 \leq 2^{\alpha}$ which implies $\alpha \geq 1$.
If $\alpha \geq 1$ consider the function $f(x)=(x+y)^{\alpha}-x^{\alpha}-y^{\alpha}$ for fixed $y$. Since $f'(x)=\alpha (x+y)^{\alpha -1} -\alpha x^{\alpha -1} \geq 0$ the function is increasing on $[0,\infty)$. Since $f(0)=0$ w get $f(x) \geq 0$ for all $x \geq 0$. This proves the inequality when $\alpha \geq 1$.
