Level curves of $Q(x_{1},x_{2})=\min\{2x_{1};x_{2}\}+\min\{x_1;2x_{2}\}$

Question

I started thinking of different cases for example suppose $2x_{1}<x_{2}$, then $x_{1}<2x_{2}$. So $Q(x_{1},x_{2})=2x_{1}+x_{2}=3x_{1}$, then $Q/3=x_{1}$. But I think this path is endless. How do you deal with an equation like this?

Answer 1

Notice that for $c\in\mathbb{R}$, $Q(x_1,x_2)=c$ for each of the three following cases:

1. $x_1=\dfrac{c}{3},\quad x_2\ge\dfrac{2c}{3}$
2. $x_1\ge\dfrac{2c}{3},\quad x_2=\dfrac{c}{3}$
3. $x_1+x_2=c,\quad \dfrac{c}{3}<x_1<\dfrac{2c}{3}$

This defines the level curve $Q(x_1,x_2)=c$

Here is a desmos graph of the level curve.