(a) To find the coordinates of \(M\), we need to find the minimum point of the curve \(y = (3-x)e^{-\frac{1}{3}x}\).

First, find the derivative \(\frac{dy}{dx}\) using the product rule:

\(\frac{dy}{dx} = -e^{-\frac{1}{3}x} - \frac{1}{3}(3-x)e^{-\frac{1}{3}x}\).

Set \(\frac{dy}{dx} = 0\) to find critical points:

\(-e^{-\frac{1}{3}x} - \frac{1}{3}(3-x)e^{-\frac{1}{3}x} = 0\).

Simplify and solve for \(x\):

\(x = 6\).

Substitute \(x = 6\) back into the original equation to find \(y\):

\(y = (3-6)e^{-\frac{1}{3} \times 6} = -3e^{-2}\).

Thus, the coordinates of \(M\) are \((6, -3e^{-2})\).

(b) To find the area of the shaded region, integrate the function from \(x = 0\) to \(x = 3\):

\(\int_{0}^{3} (3-x)e^{-\frac{1}{3}x} \, dx\).

Using integration by parts, we find:

\(-3e^{-\frac{1}{3}x}(3-x) + 9e^{-\frac{1}{3}x}\).

Evaluate from \(x = 0\) to \(x = 3\):

\(\left[-3e^{-1}(3-3) + 9e^{-1}\right] - \left[-3e^{0}(3-0) + 9e^{0}\right] = \frac{9}{e}\).

Thus, the area of the shaded region is \(\frac{9}{e}\).