Answer
$A=-\frac{(x-4)^2}{2}+8$
The maximum value for $A$ is obtained when $x=4$:
$A=8$
Work Step by Step
$A=-\frac{x^2}{2}+4x$ (see item (a))
$A=-\frac{1}{2}(x^2-8x)=-\frac{1}{2}(x^2-8x+16-16)$
$A=-\frac{1}{2}[x^2-2(4)x+4^2-16]$
$A=-\frac{1}{2}[(x-4)^2-16]=-\frac{(x-4)^2}{2}+8$
The maximum value for $A$ is obtained when $x-4=0$, that is, when $x=4$:
$A=-\frac{(4-4)^2}{2}+8=8$
