The problem asks for the minimum value of the function $f(x, y) = 8x + 5y$, subject to the following system of inequalities:

$x + 2y \geq 8, \quad x + y \geq 6, \quad x \geq 0, \quad y \geq 0$

This is a linear programming problem where we need to minimize $f(x, y) = 8x + 5y$ under the given constraints.

Steps to solve the problem:

1. Graph the system of inequalities: We graph the inequalities $x + 2y \geq 8$, $x + y \geq 6$, $x \geq 0$, and $y \geq 0$ on the coordinate plane. The feasible region is the area where all these inequalities overlap.

2. **Find the corner points