What is the subdifferential of the $f(x)$?

Question

$$f(x_1,x_2)=2|x_1|+3|x_2|$$

I did a plot of this function:

The subgradient should be on the red peak on the bottom?

Answer 1

$$\partial f(x_1,x_2) = 2\mathrm{Sgn}(x_1)\times3 \mathrm{Sgn}(x_2),$$ where $$\mathrm{Sgn}(y) := \begin{cases} \{-1\}, &\text{if $y<0$;}\\ [-1,1], &\text{if $y=0$;}\\ \{1\}, &\text{if $y>0$} \end{cases} $$ because $f$ is separable, the subdifferential of the absolute-value function is $\mathrm{Sgn}$, and the constant-multiple rule applies.

Now, the "peak on the bottom" is at $(x_1,x_2)=(0,0)$. At this point, we have $$\partial f(0,0)=[-2,2]\times[-3,3].$$