Find the maximum value of the objective function and where it occurs, subject to the indicated constraints.
Objective function:

Z = 4x + 3y

Constraints:

X ≥ 0
Y ≥ 0
X + y ≤ 4

A)Maximum at (4, 0): 17
B)Maximum at (0, 4): 12
C)Maximum at (0, 0): 0
D)Maximum at (4, 0): 16
E)Maximum at (0, 4): 13

Question

Find the maximum value of the objective function and where it occurs, subject to the indicated constraints. ​
Objective function:
​
Z = 4x + 3y
​
Constraints:
​
X ≥ 0
Y ≥ 0
X + y ≤ 4​
​​  ​
A)Maximum at (4, 0): 17 
B)Maximum at (0, 4): 12 
C)Maximum at (0, 0): 0 
D)Maximum at (4, 0): 16 
E)Maximum at (0, 4): 13

Quizplus · Accepted Answer

To solve this problem, we need to graph the constraints and find the feasible region.

The first constraint, x ≥ 0, means that x can only take on non-negative values. The second constraint, y ≥ 0, means that y can only take on non-negative values. The third constraint, x + y ≤ 4, represents a line with intercepts at (0,4) and (4,0).

Graphing the three constraints, we get the following feasible region:

​
11eae838_9cc5_c2a9_a6c3_dd82abd759d5_graph.png

To find the maximum value of the objective function, we now need to evaluate it at the corners of the feasible region. The corners are (0,0), (0,4), and (4,0).

When we plug in (0,0), we get Z = 0.

When we plug in (0,4), we get Z = 12.

When we plug in (4,0), we get Z = 16.

Therefore, the maximum value of the objective function is 16, and it occurs at (4,0).

The best choice is D, Maximum at (4, 0): 16.

Find the Maximum Value of the Objective Function and Where