The correct answer is: A. The extreme point of the oscillation.
A particle moving with a simple harmonic motion attains its maximum velocity when it passes through the extreme point of the oscillation. This is because the velocity of the particle is given by the equation $v = -\omega A \sin \omega t$, where $\omega$ is the angular frequency of the oscillation, $A$ is the amplitude of the oscillation, and $t$ is the time. The maximum value of the sine function is 1, so the maximum velocity of the particle occurs when $\sin \omega t = 1$. This happens when $t = \frac{\pi}{2} + 2 \pi n$, where $n$ is an integer. Therefore, the maximum velocity of the particle occurs when it passes through the extreme point of the oscillation.
The other options are incorrect because:
- Option B is incorrect because the particle has zero velocity when it passes through the mean position.
- Option C is incorrect because the particle has a velocity of zero when it passes through a point at half amplitude.
- Option D is incorrect because the particle does attain its maximum velocity at some point during the oscillation.