If S is a closed and bounded subset of whose boundary is a polyhedron (i.e., the boundary...

If S is a closed and bounded subset of whose boundary is a polyhedron (i.e., the boundary consists of polygons), then a linear function f of three variables attains its extreme values on the boundary of S. But since each of these faces is the graph of a linear function, it follows that f restricted to one of the faces is a linear function of two variables. This two-variable function, then, attains its extreme values on the boundary of the polygon. In other words, the extreme values of f occur on the edges of the polyhedron S. Use this fact to find the maximum and minimum values of the following functions on the indicated sets.

f(x, y, z)= x − y + z; A = {(x, y, z) | x ≥ 0, y ≥ 0, z ≥ 0, x + y + z ≤ l}