Extra explanation

Consider once again the example given at the definition of Minimum functions:
$$z(x,y) = \min\{3x,4y\}.$$

If we calculate the function value at $(x,y) = (1,2)$, then we obtain
$$z(x,y) = \min\{3\cdot1,4\cdot2\} = \min\{3,8\} = 3.$$
The minimum is given by $3x$.

If we calculate the function value at $(x,y) = (2,1)$, the we obtain
$$z(x,y) = \min\{3\cdot2,4\cdot1\} = \min\{6,4\} = 4.$$
The minimum is given by $4y$.

If we calculate the function value at $(x,y) = (1,0.75)$, then we obtain
$$z(x,y) = \min\{3\cdot1,4\cdot0.75\} = \min\{3,3\} = 3.$$
The minimum is given by both $3x$ and $4y$.


The corresponding graph is

The graph is given by two planes with a line of intersection. The left plane corresponds to $3x$, the right plane to $4y$. On the line of intersection the results of $3x$ and $4y$ are equal, hence,
$$\begin{align} 4y &= 3x,\\ y &= \tfrac{3}{4} x. \end{align}$$
The point $(x,y) = (1,2)$ (the black point) is on the left plane, the point $(x,y)=(2,1)$ (the blue point) is on the right plane and the point $(x,y)=(1,0.75)$ (the green point) is on the line of intersection.