Answer
It seems that the area $A$ when $x=4$ is the maximum area.
Work Step by Step
$x~~~~~~~~~~~~~~~~A=4x-\frac{x^2}{2}$
$1~~~~~~~~~~~~~~4(1)-\frac{1^2}{2}=\frac{7}{2}$
$2~~~~~~~~~~~~~~4(2)-\frac{2^2}{2}=6$
$3~~~~~~~~~~~~~~4(3)-\frac{3^2}{2}=\frac{15}{2}$
$4~~~~~~~~~~~~~~4(4)-\frac{4^2}{2}=8$
$5~~~~~~~~~~~~~~4(5)-\frac{5^2}{2}=\frac{15}{2}$
$6~~~~~~~~~~~~~~4(6)-\frac{6^2}{2}=6$
$7~~~~~~~~~~~~~~4(7)-\frac{7^2}{2}=\frac{7}{2}$
