Answer:
(-4, 2)∪(8, ∞)
Explanation:
Given a particle's position is described by x(t) = t³ -6t² -96t, you want the intervals where speed is increasing.
Speed
The speed of the particle is the magnitude of its rate of change of position.
The rate of change of position is ...
x'(t) = 3t² -12t -96 = 3(t² -4t) -96
x'(t) = 3(t -2)² -108
This describes a parabola that opens upward, with a vertex at (2, -108). It has zeros at x = 2 ± 6 = {-4, 8}.
The magnitude of the speed is shown by the blue curve in the attachment. Between t=-4 and t=8, it is the opposite of the parabola described by the above equation.
Acceleration
The rate of change of speed is the derivative of speed with respect to time. The green curve in the attachment shows the particle's rate of change of speed. Speed is increasing when the green curve is above the x-axis.
Between the point when speed is 0, at t=-4, and when it reaches a local maximum, at t=2, it is increasing. Speed is increasing again after it becomes 0 at t=8.
The intervals of increasing speed are (-4, 2) ∪ (8, ∞).
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Additional comment
We have made the distinction between speed and velocity. Velocity is the signed rate of change of position. If position is plotted on a number line increasing to the right, then velocity is positive anytime the particle is moving to the right. Velocity is increasing if acceleration is to the right (positive).
Velocity of this particle is increasing on the interval (2, ∞).