Question

Find the maximum and minimum values of the objective function f(x, y) and for what values of x and y they occur, subject to the given constraints.
f(x, y) = 2x + 6y
x ≥ 0
y ≥ 0
2x + 7y ≤ 70
8x + 4y ≤ 136

Answer 1

Given that f(x, y) is a linear function, and the constraints themselves are also linear, it follows that the vertices of the feasible region are the sites of extrema of f(x, y). So just find where each boundary line intersects with another line, and check the value of f(x, y) at each intersection.

We have

• x = 0 and y = 0   ===>   (0, 0)

• x = 0 and 2x + 7y = 70   ===>   y = 10   ===>   (0, 10)

• y = 0 and 8x + 4y = 136   ===>   x = 17   ===>   (17, 0)

• 2x + 7y = 70 and 8x + 4y = 136   ===>   (14, 6)

At these points, we respectively get

• f (0, 0) = 0

• f (0, 10) = 60

• f (17, 0) = 34

• f (14, 6) = 64

Then max f(x) = 64 at (14, 6) and min f(x) = 0 at (0, 0).