\(P=x+y+\frac{\sqrt{3}}{2}.2.\left(\frac{x}{\sqrt{3}}\right).\sqrt{1-y^2}+\frac{\sqrt{3}}{2}.2.\left(\frac{y}{\sqrt{3}}\right).\sqrt{1-x^2}\)
\(P\le x+y+\frac{\sqrt{3}}{2}\left(\frac{x^2}{3}+1-y^2\right)+\frac{\sqrt{3}}{2}\left(\frac{y}{3}+1-x^2\right)\)
\(P\le-\frac{\sqrt{3}}{3}x^2+x-\frac{\sqrt{3}}{3}y^2-y+\sqrt{3}\)
\(P\le-\frac{\sqrt{3}}{3}\left(x-\frac{\sqrt{3}}{2}\right)^2-\frac{\sqrt{3}}{3}\left(y-\frac{\sqrt{3}}{2}\right)^2+\frac{3\sqrt{3}}{2}\le\frac{3\sqrt{3}}{2}\)
\(P_{max}=\frac{3\sqrt{3}}{2}\) khi \(x=y=\frac{\sqrt{3}}{2}\)