Question

using the geometric approach, we want to minimize the objective function (z = 4x + 5y) subject to the following constraints: (\begin{cases}x + yleq6\\y - xgeq2\xgeq0\ygeq0end{cases}). the graph below shows the boundary lines for the above constraint inequalities but is not shaded. select the one answer that contains the correct coordinates of the corner points of what must be shaded as the solution - region, and identify the associated minimum value given the above objective function, (z).

Explanation:

Step1: Identify boundary lines intersections

Solve \(x + y = 6\) and \(y = x + 2\): \(x=2\), \(y=4\) → (2,4).

Step2: Find intercepts with axes

\(y = x + 2\) at \(x=0\): (0,2); \(x + y = 6\) at \(x=0\): (0,6).

Step3: Determine corner points

Feasible region corners: (0,2), (0,6), (2,4).

Step4: Calculate Z at each corner

At (0,2): \(Z=4(0)+5(2)=10\); (0,6): \(Z=30\); (2,4): \(Z=28\).

Answer:

Corner points: (0, 2), (0, 6), (2, 4); Minimum Z=10