A feasible region has vertices at (-3,2),(4,1),(2,6) and (1,-2). Find the maximum an minimum values of the function f(x,y)=4x-y
To find the maximum and minimum values of the function f(x,y) = 4x - y within the feasible region, we need to evaluate the function at each of the vertices and compare the results.
1. Start by plugging in the coordinates of the first vertex, (-3, 2), into the function:
f(-3, 2) = 4(-3) - 2 = -12 - 2 = -14
2. Next, evaluate the function at the coordinates of the second vertex, (4, 1):
f(4, 1) = 4(4) - 1 = 16 - 1 = 15
3. Then, plug in the third vertex, (2, 6):
f(2, 6) = 4(2) - 6 = 8 - 6 = 2
4. Finally, evaluate the function at the last vertex, (1, -2):
f(1, -2) = 4(1) - (-2) = 4 + 2 = 6
By comparing the function values, we can determine that the maximum value within the feasible region is 15 (at vertex (4, 1)) and the minimum value is -14 (at vertex (-3, 2)).
To find the maximum and minimum values of the function f(x, y) = 4x - y in the given feasible region, we need to evaluate this function at each vertex and compare the values.
Step 1: Evaluate the function at each vertex:
At (x, y) = (-3, 2):
f(-3, 2) = 4(-3) - 2 = -12 - 2 = -14
At (x, y) = (4, 1):
f(4, 1) = 4(4) - 1 = 16 - 1 = 15
At (x, y) = (2, 6):
f(2, 6) = 4(2) - 6 = 8 - 6 = 2
At (x, y) = (1, -2):
f(1, -2) = 4(1) - (-2) = 4 + 2 = 6
Step 2: Compare the values to find the maximum and minimum.
The maximum value is 15, which occurs at (4, 1).
The minimum value is -14, which occurs at (-3, 2).
Therefore, the maximum value of the function f(x, y) = 4x - y in the given feasible region is 15, and the minimum value is -14.