showing $y\to |y|^{p}$ is convex $p\geq 1$

$y\to |y|^{p}$ is convex only for $p\geq 1$ and $y\in \mathbb{R}$.

This function is nondifferentiable but we can see that the second derivative is nonnegative in each interval $(-\infty,0)$ and $(0,\infty)$. So each wing is convex.

Can I use this to prove convexity for the original function? If not can you give a hint?Thanks