What is a convex function? State Jensen's inequality for a convex function of a random variable which takes finitely many values.

Let $p \geqslant 1$. By using Jensen's inequality, or otherwise, find the smallest constant $c_{p}$ so that

$$(a+b)^{p} \leqslant c_{p}\left(a^{p}+b^{p}\right) \text { for all } a, b \geqslant 0 .$$

[You may assume that $x \mapsto|x|^{p}$ is convex for $p \geqslant 1$.]