Question

For the feasibility region shown below, find the maximum value of the function P=2x+3y.

Answer 1

Let

A--------> the corner point in the graph $(0,0)$

B--------> the corner point in the graph $(0,8)$

C--------> the corner point in the graph $(6,5)$

D--------> the corner point in the graph $(8,0)$

we know that

The function P is equal to

$P=2x+3y$

Step $1$

Evaluate the function P in each corner point

point $A(0,0)$

$x=0\\ y=0\\ PA=2*0+3*0\\ PA=0$

point $B(0,8)$

$x=0\\ y=8\\ PB=2*0+3*8\\ PB=24$

point $C(6,5)$

$x=6\\ y=5\\ PC=2*6+3*5\\ PC=27$

point $D(8,0)$

$x=8\\ y=0\\ PD=2*8+3*0\\ PD=16$

the maximum value of P is for the point C

$PC=27$

therefore

the answer is

The maximum value of the function P is $27$

Answer 2

Corner points in this graph are: ( 0,0 ) ( 0,8 ) ( 5,6 ) and ( 8, 0 ).
If we plug those values in : P = 2 x + 3 y
P ( 0,0 )= 0
P ( 0,8 ) = 2 * 0 + 3 * 8 = 24
P ( 6 , 5 ) = 2 * 6 + 3 * 5 = 12 + 15 = 27
P ( 8 , 0 ) =  2 * 8 + 3 * 0 = 16
The maximum value is:
P max ( 6 , 5 ) = 27