how Determine the maximum values of C. how Determine the maximum values of C.

my try is that :
To graph the last two bounding lines, I'll want to put the equations into slope–intercept form.
The bounding line corresponding to the 3rd constraint inequality becomes:
2y = 8 + 4x
y = 4 + 2x
And the bounding line corresponding to the 4th constraint inequality becomes:
y = 4 – 2x
I need to know how get From the graph the vertices of the region to get the the maximum values of C.
 A: When working on Linear programming problems, we use a consistent approach by attempting to:


*

*Evaluate the objective function $C$ for each labeled point in the feasible region

*Pick points on the region endpoints and points inside the region. 

*At which labeled point does the maximum value of $C$ occur? At which labeled point
does the minimum value of $C$ occur?

*What are the maximum and minimum values of $C$ on the entire feasible region? 

*Try other constraints using points in the region to see if you can find values $x$ of $C$ that are greater or lesser than those you $y$ found.


Lets try this approach on your problem. If we draw the first two constraints ($0 \le x \le 6, 0 \le y \le 6$), we have the feasible region:

If we then add the last two constraints, the feasible region (the top left cutout is from $y \le 2x + 4$ and the bottom left cutout is from $y \ge -2x + 4$) is:

Now, we want to maximize:
$$C(x, y) = 3x + y$$
We can test each of the vertices as:


*

*$C(2, 0) = 6$

*$C(6, 0) = 18$

*$C(0, 4) = 4$

*$C(1, 6) = 9$

*$C(6, 6) = 24$

*Per the approach described above, we can also test points inside the region and see that they will not provide a maximum.


It is clear which is the maximum.
We could have also noted that the function $3x + y$ is increasing for positive $x$ and $y$ and from the feasible region plot, the max is given by $(x, y) = (6, 6)$. The same technique can be used to determine the minimum point.
