Find the maximum and minimum values of the function for this region

Question

x ≥ -1

y ≤ -x + 6

f(x,y) = x + 2y

I know this problem is unbounded but I get confused how to decide if its min or max.

I tried it and got:

Min: -1 at (-1,0)

is this correct?

Answer 1

Let's start with x = -1 (because we are given that x > -1)

When x = -1, then y ≤ -(-1) + 6 or y ≤ 7 That means f(x, y) will be 13 or smaller (because f(x, y) = x + 2y)

Now let's consider x = 0 (one integer up from -1). When x = 0, y ≤ 0 + 6 or y ≤ 6. That means f(x,y) will be 12 or smaller.

Now, let's consider x = 1. If x = 1, then y ≤ -(1) + 6 or y ≤ 5 making f(x,y) equal to 11 or smaller.

And, lastly, let x = 2. If x = 2, then y ≤ -(2) + 6 or y ≤ 4 making f(x,y) equal to 10 or smaller.

And the relationship keeps going.

So the MAXIMUM occurs at x = -1 and the value is 13. And, like you mentioned, the function is unbounded but on the negative side (BECAUSE of the "negative x" + 6) so there is no minimum value.