Proof that $x^\frac{1}{p}$ is concave without derivative $(p>1)$

Question

I need to show that $f(x_1,...,x_n)= \sum_{i=1}^{n} x_i^\frac{1}{p}$ is concave (for $x_i>0$ and $p>1$) without using any derivatives.

I have attempted to prove that the secant line lies below the curve for the one-variable function $x \in R_+^*\mapsto x^\frac{1}{p}$ but am stuck.

Any suggestions would be much appreciated !