The diagram shows a sketch of the curve \(y = x^{\frac{1}{2}} \ln x\) and its minimum point \(M\). The curve cuts the \(x\)-axis at the point \((1, 0)\).

(i) Find the exact value of the \(x\)-coordinate of \(M\).

(ii) Use integration by parts to find the area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = 4\). Give your answer correct to 2 decimal places.