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Find out the free coordinates designated by $D(x,y)$ in the two$-$dimensional Euclidean space which rests on the function $y=\sqrt[\phantom{2}]{x}$ that maximizes the area S which is formed by the partially or completely fixed points $A(x,0),B(3,0),C(3,y)$ and the free point $D(x,y).$

Calculate optimum value $S^{*}.$

Note: the value of the x co$-$cordinate of $D(x,y)$ is less than $3.$ Use Lagrange multiplier technique