Find the maximum value of the objective function and the values of x and y for which it occurs.

Question

F = 5x + 2y
x + 2y (greater than or equal to) 6
2x + y (greater than or equal to) 6
Both x and y are greater than or equal to 0.

I don't understand how to do this! And I can't remember how to graph those functions. It's been a while. :-/ Any help would be GREATLY appreciated! Thank you in advance!!!

Answer 1

This is a problem is what is called linear programming.

I have a sneaking suspicion that the objective function F = 5x+2 should have a minimum value.
e.g. I could pick a huge x and a huge y
say, x=500, y = 800
then F = 2500 + 1600 = 4100
and (500,800) satisfy both of the inequations
I could get a "larger" value of F by increasing my x's and y's.
So F has no maximum.
sketch x+2y ≥ 6

So let's assume you meant to find a Minimum of F

Now to your question:
the simplest way is to graph x + 2y = 6 and shade in the region above that line including the line.
I would just mentally calculate the x and y intercepts to get (0,3) and (6,0)

do the same thing for 2x + y ≥ 6

Now shade in the region that belongs to both x+2y ≥ 6 and 2x+y≥6

It is easy to see that they intersect at (2,2)

So you have 3 critical values
(0,6) (2,2) and (6,0)
which of these gives the smallest value of F ?
try (2,2)
F = 5(2) + 2(2) = 14
for (0,6)
F = 5(0)+2(6) = 12
for (6,0)
F = 5(6) + 2(0) = 30

So what do you think?
Check my arithmetic, I tend to make silly errors lately.