Find the Maximum and minimum value of the function for this region

Question

y ≥ 0

x ≥ 0

y ≤ -x + 10

y ≤ 2x +1

f (x,y) = 5x + 2y

I have it all graphed out but I'm unsure if I'm supposed to do substitution for using the x ≥ 0 and y ≤ 2x +1

to get that vertical point.

Answer 1

So the graph looks like this:

Things happen at the vertices. So lets's find f(x,y) at each:

f(0,0) = 5(0) + 2(0) = 0

f(0,1) = 5(0) + 2(1) = 2

f(3,7) = 5(3) + 2(7) = 29

f(10,0) = 5(10) + 2(0) = 50

Just to make sure we believe the fact that things happen at the vertices, let's try a few other points:

f(2,5) = 5(2) + 2(5) = 20

f(6,4) = 5(6) + 2(4) = 38

f(9,1) = 5(9) + 2(1) = 47

So it appears that our "vertex numbers" are good. Our max is found at x = 10, y = 0 and the maximum value is 50 and our minimum is found at x = 0 and y = 0 and the minimum value is 0.