(i) To find the \(x\)-coordinate of the maximum point \(M\), we need to find the derivative of \(y = (2x - x^2)e^{\frac{1}{2}x}\) and set it to zero.

Using the product rule, the derivative is:

\(\frac{d}{dx}[(2x - x^2)e^{\frac{1}{2}x}] = (2 - 2x)e^{\frac{1}{2}x} + \frac{1}{2}(2x - x^2)e^{\frac{1}{2}x}\)

Setting the derivative to zero:

\((2 - 2x)e^{\frac{1}{2}x} + \frac{1}{2}(2x - x^2)e^{\frac{1}{2}x} = 0\)

Simplifying and solving for \(x\), we find:

\(x = \sqrt{5} - 1\)

(ii) To find the area of the shaded region, we need to integrate \(y = (2x - x^2)e^{\frac{1}{2}x}\) from \(x = 0\) to \(x = 2\).

Using integration by parts, we have:

\(\int (2x - x^2)e^{\frac{1}{2}x} \, dx\)

Integrating by parts twice, we obtain:

\((12x - 2x^2 - 24)e^{\frac{1}{2}x}\)

Evaluating from \(x = 0\) to \(x = 2\):

\([12(2) - 2(2)^2 - 24]e^{1} - [12(0) - 2(0)^2 - 24]e^{0}\)

\(= 24e - 8e - 24\)

\(= 24 - 8e\)