$\left(x-\frac 12 \right)+3\left(y-\frac 12 \right) \overset{Cauchy-Schwarz}\le \sqrt{(1^2+3^2)\left[ \left(x-\frac 12 \right)^2+\left( x-\frac 12 \right) \right]^2}$
$\le \sqrt{10.\frac 12}=\sqrt 5$
$\Leftrightarrow x+3y-2 \le \sqrt 5$
$\Rightarrow \max A=2+\sqrt5$ khi $\begin{cases}x=\dfrac {5+\sqrt 5}{10} \\ y=\dfrac{5+3\sqrt5}{10} \end{cases}$