subdifferential of $\max_{i=1,\cdots,k} x_i+\frac{1}{2}\|x\|_2^2,\ \ \ x\in \mathbb{R}^n$

Question

How to find the subdifferential of
$$f(x) = \max_{i=1,\cdots,k} x_i+\frac{1}{2}\|x\|_2^2,\ \ \ x\in \mathbb{R}^n$$

My derivation is:

1. $\nabla \frac{1}{2}\|x\|_2^2=\nabla \frac{1}{2}x^Tx=x$

2. The solution for the $\nabla \max_{i=1,\cdots,k} x_i=\text{conv} \{e_i\ \ \big | \ \ i= \arg \max_j x_j\}$, where $e_i$ is the coordinate vector of $i$-th coordinate.

I am a little bit shaky about the second one. Particularly, my problem is on why is $e_i$ and why take convex hull? Could anyone give me an intuitive and simple way to think of this? or an example.