How to show that $(a+b)^p\le 2^p (a^p+b^p)$ If I may ask, how can we derive that $$(a+b)^p\le 2^p (a^p+b^p)$$ where $a,b,p\ge 0$ is an integer? 

Using Jensen's Inequality, we get for $p\ge1$ or $p\le0$,
$$
\left(\frac{A+B}2\right)^p\le\frac{A^p+B^p}2
$$
Which, upon multiplication by $2^p$, yields
$$
(A+B)^p\le2^{p-1}(A^p+B^p)
$$
