Solving without options:
f(x,y)=3X+6Y
max of f(x,y) will be obtained when both X and Y are having their max values.
2X+Y≤6 --> (1) max(2X+Y)=6
X+2Y≤8 -->(2) max(X+2Y)=8
Adding (1) and (2),
3X+3Y≤14
=> X+Y≤14/3 => max(X+Y)=14/3 ---> (3)
Max of f(X,Y)= max(3X+6Y)= max (3(X+2Y)) = 3*max(X+2Y) =3*8=24 ---> (4)
Max of f(X,Y)= max(3X+6Y)=max(3(X+Y) + 3Y) = 3*max(X+Y) + 3*max(Y) = 3*14/3 + 3*max(Y) --->(5)
Equating (4) and (5),
24=14+3*max(Y)
=> max(Y)=10/3
Placing Y in Eq (3),
max(X)= 14/3-10/3=4/3