First, find the first derivative \(f'(x)\):

\(f'(x) = \frac{d}{dx}[2x + (x + 1)^{-2}] = 2 - 2(x + 1)^{-3}\).

Next, find the second derivative \(f''(x)\):

\(f''(x) = \frac{d}{dx}[-2(x + 1)^{-3}] = 6(x + 1)^{-4}\).

Evaluate \(f'(x)\) at \(x = 0\):

\(f'(0) = 2 - 2(1)^{-3} = 0\).

This shows that \(x = 0\) is a stationary point.

Evaluate \(f''(x)\) at \(x = 0\):

\(f''(0) = 6(1)^{-4} = 6 > 0\).

Since \(f''(0) > 0\), the function has a minimum at \(x = 0\).