Final answer:
The speed of the particle is decreasing in the intervals (-∞, 3) and (3, ∞).
Step-by-step explanation:
The speed of a particle can be determined by taking the derivative of its position function.
In this case, the position function is given by x(t) = t^3 - 9t^2 - 81t.
To find the speed function, we take the derivative of x with respect to t.
v(t) = dx/dt = 3t^2 - 18t - 81
The speed is decreasing when the derivative of the speed function is negative. To find the intervals when the speed is decreasing, we take the derivative of v with respect to t.
a(t) = dv/dt = 6t - 18
To determine the intervals when the speed is decreasing, we set the derivative equal to zero and solve for t.
6t - 18 = 0
t = 3
Since the acceleration is positive for t < 3 and negative for t > 3, the speed is decreasing in the intervals (-∞, 3) and (3, ∞).