Final answer:
To find when the particle reaches minimum velocity, we differentiate the position function to find velocity, then differentiate again to find acceleration and set it to zero. Solving for time, we get approximately 0.78 seconds, but since this isn't a provided option, the closest would-be t=1 s.
Step-by-step explanation:
The student has asked when the particle reaches its minimum velocity as given by the position function x = (3t^3 - 7t^2 + 12) meters. To find the time when the particle reaches minimum velocity, we first need to find the velocity function by differentiating the position function with respect to time. This yields the velocity function v(t) = dx/dt = 9t^2 - 14t. To find when the velocity is at a minimum, we set the derivative of the velocity function (which is the acceleration function) to zero, so d/dt (v(t)) = d/dt (9t^2 - 14t) = 18t - 14. Setting 18t - 14 = 0 gives us t = 14/18, or approximately t = 0.78 seconds, which is not one of the offered answers. It appears there may be a mistake in the question as posed since none of the answer choices match the correct calculation. However, if we had to choose the closest provided option, it would be t=1 s.