$A = B + C$
For $A$ to be maximum, both $B$ and $C$ should be maximum
$B = D - E$
For $B$ to be maximum, $D$ should be maximum and $E$ should be minimum
$C = F + G$
For $C$ to be maximum, both $F$ and $G$ should be maximum
The maximum value of $D$ is $2\;( 1 + 1 = 2 )$
The minimum value of $E$ is $-1 \;( 0 - 1 = -1 )$
The maximum value of $F$ is $1 \;( 1 - 0 = 1 )$
The maximum value of $G$ is $2 \;( 1 + 1 = 2 )$
$B = 2 - ( -1 ) = 2 + 1 = 3$
$C = 1 + 2 = 3$
$A = B + C = 3 + 3 = 6$
$6$ is the answer