Answer:
Hi,
Maximum=40
Answer C
Explanation:
To find the maximum value of the function f(x, y) = 8x - 3y subject to the given constraints, you don't need to use Lagrange multipliers because the constraints are simple linear inequalities. Instead, you can graphically determine the solution.
The constraints are as follows:
0 ≤ y ≤ 5
y ≤ -x + 5
y ≤ x + 5
First, consider constraint 1. It states that 0 ≤ y ≤ 5, which means the values of y are bounded between 0 and 5. Now, let's plot this constraint in a coordinate system.
Graph the line y = 0. This represents y = 0.
Graph the line y = 5. This represents y = 5.
Shade the region between these two lines.
Next, consider constraints 2 and 3:
y ≤ -x + 5
y ≤ x + 5
Plot these two lines as well:
For constraint 2, plot the line y = -x + 5.
For constraint 3, plot the line y = x + 5.
Now, let's look at the shaded region that satisfies all three constraints:
The feasible region is the area where all three shaded regions overlap. This region is a triangle with vertices at (0, 0), (5, 0), and (0, 5).
To find the maximum value of f(x, y) = 8x - 3y within this feasible region, you can evaluate the function at each vertex of the triangle:
f(0, 0) = 8(0) - 3(0) = 0
f(5, 0) = 8(5) - 3(0) = 40
f(0, 5) = 8(0) - 3(5) = -15
The maximum value of the function within the feasible region is 40, which occurs at the point (5, 0). Therefore, the maximum value of the function f(x, y) = 8x - 3y subject to the given constraints is 40, and it is achieved when x = 5 and y = 0.