Question

What point in the feasible region maximizes the objective function objective function c=5x-4y?

Answer 1

Hat point in the feasible region maximizes the objective function 
constraints: 
x>=0 
y>=0 
-x+3>=y 
y<=1/3 x+1 

Objective function: C=5x-4y 

1. Region limited by : 
x>=0 
y>=0 
x + y <= 3 
is the interior of rectangle triangle 
of summits (0,0), (0,3)and (3,0) 
if we add the constraint 
y <= 1/3 x + 1 
it's the part in the triangle below this line : 
the summits are (0,0) , (0,1) , (3,0) 
and the intersection point of 
line L of equation : y = x/3 + 1 and the hypotenuse 
of the triangle (equation x+y = 3) 
let's solve this : 
3 - x = x/3 + 1 
4x/3 = 2 
x = 3/2 and y = 3/2 

now the Criteria : C = 5x - 4y 
are lines parallel to line of equation 
5x - 4y = 0 
or 
y = (5/4)x 

so C is maximum at an edge of the domain : 
points are 
O ( 0 ,0) 
A( 3 , 0) 
B ( 0 ; 1) 
D ( 3/2 ; 3/2) 

criteria is C = 5x - 4y 
C (A) = 5*3 - 4*0 = 15 
C(B) = 5*0 - 4*1 
C(D) = 5* (3/2) - 4*(3/2) = 3/2 

so C is max at point A(3 ; 0)